Questions on Algebra: Exponent and logarithm as functions of power answered by real tutors!

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Tutors Answer Your Questions about Exponential-and-logarithmic-functions (FREE)


Question 571504: how to solve
299^99

Answer by Alan3354(21554) About Me  (Show Source):
You can put this solution on YOUR website!
how to solve
299^99
=================
There's nothing to solve. Equations are solved, this is not one.
----
The result of the calculation is a big number.
It's 1.23437649...E245, 240+ digits


Question 571114: Write an exponential function for a graph that includes point (1, 0.84) and (2,1.008). I've gotten as far as 0.84/b=a , and 1.008=(.84/b)*b^2, but I'm stuck there.
Answer by KMST(578) About Me  (Show Source):
You can put this solution on YOUR website!
I would say the function is y=a%2Ab%5Ex first.
Then, I would substitute the coordinates of the points given and look at what I get for ideas.
0.84=ab
1.008=ab%5E2
Once you look at those two equations, you may think of a good way to solve it, that would not give you much trouble. Even if you do not choose the easiest or fastest way to the solution, do not panic. Try to stay calm and confident.
ONE OPTION
You could decide on dividing the second equation by the first one
1.008%2F0.84=ab%5E2%2Fab --> 1.008%2F0.84=b or b=1.008%2F0.84 --> b=1.2
Once you get b, you can substitute in 0.84=ab to find a.
0.84=a%2A1.2 --> 0.84%2F1.2=a or a=0.84%2F1.2 --> a=0.7
So the function is highlight%28y=0.7%2A1.2%5Ex%29
ANOTHER OPTION
You could decide on solving for a in the first equation and substituting that into the second one.
For example, you found that 0.84%2Fb=a and substituted to get
1.008=%280.84%2Fb%29%2Ab%5E2
That may look scary, but can be simplified
1.008=%280.84%2Fb%29%2Ab%5E2 --> 1.008=0.84%2Fb%2Ab%5E2 --> 1.008=0.84b%2Ab%2Fb --> 1.008=0.84b%2A%28b%2Fb%29 --> 1.008=0.84b --> b=1.008%2F0.84 b=1.2
amd you can substitute that into 0.84=ab to find a, as shown above.
This option may look more complicated, but still leads to the solution.
AND YET ANOTHER OPTION
You could decide on solving forb in the first equation and substituting that into the second one.
b=0.84%2Fa substituted into 1.008=ab%5E2
1.008=a%2A%280.84%2Fa%29%5E2 --> 1.008=a%2A%280.84%5E2%2Fa%5E2%29 --> 1.008=a%2A0.84%5E2%2Fa%5E2 --> 1.008=a%2A0.84%5E2%2F%28a%2Aa%29 --> 1.008=%28a%2Fa%29%2A%280.84%5E2%2Fa%29 --> 1.008=0.84%5E2%2Fa --> 1.008a=0.84%5E2 --> a=0.84%5E2%2F1.008 --> a=0.7
and you can substitute that into 0.84=ab to find b:
0.84=0.7b --> 0.84%2F0.7=b or b=0.84%2F0.7 --> b=1.2
COMMENT
Staying calm is difficult, but necessary. Being confident is even harder. Having seriously practiced algebra in previous years will help give you confidence, and make calculation mistakes less likely. If you did not take algebra that seriously before, I can tell you that I've been there, and that there are plenty of people in the same boat. Several times, I found out the hard way that math, like languages and music, gets much harder the longer that you do not practice it seriously and regularly. Fortunately, at various points, I stumbled into very studious study buddies that made me practice math a lot more and be (temporarily) a little more responsible.


Question 571058: show that log8 1000=log2 10 algebraically
Answer by stanbon(48516) About Me  (Show Source):
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show that log8 1000=log2 10 algebraically
---
log(1000)/log(8) = log(10)/log(2)
--------------------
log(1000)/log(10) = log(8)/log(2)
-----
log10(1000) = log2(8)
-----
3 = 3
====================
Cheers,
Stan H.
====================


Question 571057: rewrite equation in logarithmic form
64 1/2 = 8

Answer by stanbon(48516) About Me  (Show Source):
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rewrite equation in logarithmic form
64^(1/2) = 8
---------------
base = 64
exponent = log = (1/2)
result = 8
------
Log Form:
log64(8) = 1/2
===================
Cheers,
Stan H.
===================


Question 571056: rewrite equation in exponential form log u 15/16 = v
Answer by nyc_function(2626) About Me  (Show Source):
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Use the fact that log_b (x) = y can be written b^y = x.

u^v = (15/16)


Question 570869: solve each equation. Round to four decimal places.
2^n=sqrt(3^n-2)

Answer by Alan3354(21554) About Me  (Show Source):
You can put this solution on YOUR website!
2%5En=sqrt%283%5E%28n-2%29%29
Square both sides
2%5E%282n%29+=+3%5E%28n-2%29 assuming n-2 is the exponent. Use parentheses.
2n*log(2) = (n-2)*log(3) = nlog(3) - log(9)
2n*log(2) - n*log(3) = -log(9)
n*(log(3) - 2log(2)) = log(9)
n = log(9)/log(3/4)
n =~ -7.6377


Question 569183: Please help!!I have been working on these for hours.
Solve each equation:
a. ln(2x-1)=3
b. log x = 1/10,000
c. ln(2x-1) = 3
d. ln(x+3) = 1
Thank you so much!!

Answer by Tatiana_Stebko(1060) About Me  (Show Source):
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a. ln%282x-1%29=3
If ln%28a%29=b, then a=e%5Eb
2x-1=e%5E3 add 1 to the both sides of the equation
2x=e%5E3%2B1 divide by 2
x=%28e%5E3%2B1%29%2F2
use scientific calculator x=10.54
b. log+x+=+1%2F10000
If log%28a%29=b, then a=10%5Eb
x+=+10%5E%281%2F10000%29
d. ln%28x%2B3%29+=+1+
x%2B3=e%5E1
x=e-3


Question 568663: I'm not entirely sure this is the right subject, sorry if it's not...
A substance decomposes radioactively. Its half-life is 32 years. Find the constant "k" in the decay formula.
Use this formula instead: y=ne%5Ekt
y= final value
n= initial value
k= constant
t= time
e= antilogarithm

Answer by solver91311(12118) About Me  (Show Source):
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is NOT the antilogarithm. is the base of the natural logs, that is:

If one half-life has elapsed, then .

Since the half-life of the substance in question is 32 years, then



Take the natural log of both sides:



Use



to write



Use "the difference of the logs is the log of the quotient",



to write



Use



to write



then use



to write



then



John

My calculator said it, I believe it, that settles it
The Out Campaign: Scarlet Letter of Atheism



Question 567783: solve for x: 8ln(1/2x)=72
Answer by rapaljer(4551) About Me  (Show Source):
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Let me start by saying, I don't know if the problem is
8+ln%281%2F%282x%29%29=72 or 8+ln%28%281%2F2%29x%29=+72 I will solve it both ways!

Solving the SECOND statement above:
8+ln+%28x%2F2%29+=+72

Divide both sides by 8:
ln%28x%2F2%29=9

Raise both sides as a power of e:
e%5E%28ln%28x%2F2%29%29=e%5E9
x%2F2=e%5E9

Multiply both sides by 2:
x=2e%5E9

Solving the FIRST statement above:
8 ln(1/2x) = 72

Divide both sides by 8:
ln(1/2x) = 9

Raise both sides as a power of e:
e%5E%28ln%281%2F2x%29%29=e%5E9
1%2F2x=e%5E9%2F1

Cross multiply to get this:
2xe%5E9=1
x=+1%2F%282e%5E9%29


For a complete explanation of this and other topics in LOGARITHMS, please see my own website. The best way to find it is to use the easy-to-remember and easy-to-spell link www.mathinlivingcolor.com. At the bottom of the page there is a single link that takes you to my Homepage. On my Homepage, look for the link "Basic, Intermediate, and College Algebra: One Step at a Time." Choose "College Algebra", and look in "Chapter 4" for a complete explanation of logarithms. Here you will find my own non-traditional explanation that my own students used to find much easier to understand than most of the other textbooks being used. I also have lots of examples, exercises, and answers. In addition, most of the hardest problems are solved in color on the MATH IN LIVING COLOR pages that go with this topic. There is TWO VIDEOS of me teaching LOGARITHMS in class before I retired a few years ago. To find the videos, look on my Homepage for "Rapalje Videos in Living Color," choose "College Algebra" and select the "Logarithms" videos. It's ALL FREE on my website!!!

To contact me please use my Email address at rapaljer@seminolestate.edu.

Dr. Robert J. Rapalje, Retired
Seminole State College of Florida
Altamonte Springs Campus


Question 567785: Given logk 3=7, logk 4=13, and logk 5=22, find logk (12/5k^3)^2
Answer by solver91311(12118) About Me  (Show Source):
You can put this solution on YOUR website!


Use

The sum of the logs is the log of the product, the difference of the logs is the log of the quotient, and the log of something to a power is the power times the log:







to write:



Then substitute the values you were given plus the fact that to write:



And do the arithmetic.

John

My calculator said it, I believe it, that settles it
The Out Campaign: Scarlet Letter of Atheism



Question 567762: solve for x: log2(x+2) + log2(x+6)=5
Answer by lwsshak3(2915) About Me  (Show Source):
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solve for x:
log2(x+2) + log2(x+6)=5
log2[(x+2)(x+6)]=5
convert to exponential form: base(2) raised to log of number(5)=number(x+2)(x+6)
2^5=x^2+8x+12
32=x^2+8x+12
x^2+8x-20=0
(x+10)(x-2)=0
x=-10(reject, x>0)
or
x=2


Question 566764: Natural log base of e to the -4
Answer by solver91311(12118) About Me  (Show Source):
You can put this solution on YOUR website!


I presume you mean



Use



to write



Then use



to write



John

My calculator said it, I believe it, that settles it
The Out Campaign: Scarlet Letter of Atheism



Question 566524: find r if 0.1145=r cubed

Answer by rfer(10417) About Me  (Show Source):

Question 566148: I am having difficulty comprehending how this is done. Please Help!
Use the formula v= 4/3 pi r^3
1. If the radius of a ball is twice that of another, how many times as big is its volume?
2. If the volume is twice that if a blue ball, how many times as big is its radius?

Thank you
Is

Answer by Alan3354(21554) About Me  (Show Source):
You can put this solution on YOUR website!
Use the formula v= 4/3 pi r^3
1. If the radius of a ball is twice that of another, how many times as big is its volume?
8 times.
V1+=+4%2Api%2Ar%5E3%2F3
V2+=+4%2Api%2A%282r%29%5E3%2F3+=+4%2Api%2A8r%5E3%2F2
The volume is related to r^3. (n*r)^3 = n^3*r^3
Volume is a function of the cube of the radius.
--------------------
2. If the volume is twice that if a blue ball, how many times as big is its radius?
Volume is a function of the cube of the radius.
I don't know about the color, but the radius is root%283%2C2%29 times.


Question 566011: Potassium 42 has a decay rate of approximately 5.5% per hour, assuming an exponential decay model, fine the number of hours it will take for the original quantity of potassium 42 to b halved?
Found 2 solutions by KMST, stanbon:
Answer by KMST(578) About Me  (Show Source):
You can put this solution on YOUR website!
The decay function could be represented as
fraction remaining remaining=e%5E%28-kt%29
With t in hours, we know that at t=1, fraction remaining=1-0.055=0.945
0.945=e%5E%28-k%29 <---> ln%280.945%29=-k
We want to find t so that 50% is left, so
0.5=e%5E%28-kt%29 <---> ln%280.5%29=-kt
Combining both equations
ln%280.5%29%2Fln%280.945%29=t
t=12.25

Answer by stanbon(48516) About Me  (Show Source):
You can put this solution on YOUR website!
Potassium 42 has a decay rate of approximately 5.5% per hour, assuming an exponential decay model, find the number of hours it will take for the original quantity of potassium 42 to b halved?
----
y = ab^x
---
(1/2)a = a*b^x
---
1/2 = b^x
---
1/2 = 0.945^x
x = log(1/2)/log(0.945)
--
x = 12.25 years
=====================
Cheers,
Stan H.
================


Question 566009: potassium 42 has a decay rate of approxmiately 5.5% per hour, assuming an exponential decay model, find the number of hours it will take for the original quantity of potassium 42 to be halved?
Answer by stanbon(48516) About Me  (Show Source):

Question 467405: A grandmother is looking for a plan to finance her new grandchild’s college education. She has $25,000 to invest. Search the internet and locate a long-range investment plan, CD, Savings Bond, etc, for the grandmother. The plan is to earn compound interest.
Calculate the future value of the investment. You must use the advertised interest rate,which is 3.40% the number of compounding periods per year,which and the time the funds will be invested. If you are not given the number of compounding periods a year, make it up.
The principal is $25,000. This is P.
Research the annual interest rate for your investment. This is r.
State the time in years for the investment (as in when the new grandchild will be attending college) which is 18. This is t.
State the number of compounding periods per year.2, This is n.
Model the future value of Grandma’s investment as an exponential function, with time as the independent variable: F(t) = P(1 + r/n) nt
State the future value of Grandma’s investment.
Use the internet or library resources to find the average cost of a college education today; will grandma’s investment be able to cover the cost in today’s dollars; what about in the future?
Summarize your findings in writing using proper style and grammar.

Answer by deeakram(1) About Me  (Show Source):

Question 565892: if log(x+1) + log(x-1) = 5, then what expression is equal to 10^5
Answer by stanbon(48516) About Me  (Show Source):
You can put this solution on YOUR website!
if log(x+1) + log(x-1) = 5, then what expression is equal to 10^5
-----
log[(x+1)(x-1)] = 5
---
log[x^2-1] = 5
----
x^2-1 = 10^5
====================
Cheers,
Stan H.
====================


Question 565529: Please Help with this. My teacher did not explain how to do these problems.
Use the formula D^3 = 216T^2
1. Find what happens to the duration of a storm if its diameter is doubled.
2. determine how long a storm will last if it has a diameter of 10 mi.
Thank you so much.

Answer by stanbon(48516) About Me  (Show Source):
You can put this solution on YOUR website!
Use the formula D^3 = 216T^2
1. Find what happens to the duration of a storm if its diameter is doubled.
T = sqrt[D^3/216]
-----------------------
Find "T" when D is doubled:
T = sqrt[(2D)^3/216]
T = sqrt(8D^3/216]
T = (2sqrt(2)


Question 565426: how do i solve rational exponents and evaluate like 16 -3/2
Answer by Alan3354(21554) About Me  (Show Source):
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how do i solve rational exponents and evaluate like 16 -3/2
-------
16+-+3%2F2+=+29%2F2


Question 565053: how do i solve and check
4^2x=32^1/2

Answer by ad_alta(170) About Me  (Show Source):
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Solve:
1) (4^2)x=(32)^(1/2)
2) 16x=4*sqrt(2)
3) x=sqrt(2)/4
Check:
1) 16*(sqrt(2)/4)=4*sqrt(2)
2) Also, (2*16)^(1/2)=4*sqrt(2)


Question 564990: Someone bought a CD player for $250 and finds 2 years later when he wants to sell it that it's worth $100. Find a formula for the value of the CD player if the value decreases.
Answer by stanbon(48516) About Me  (Show Source):
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Someone bought a CD player for $250 and finds 2 years later when he wants to sell it that it's worth $100. Find a formula for the value of the CD player if the value decreases.
============
You have 2 points: (0,250) and (2,100)
slope = (100-250)/(2-0) = -150/2 = -75
------
Form: y = mz + b
250 = -75*0 + b
b = 250
------
Equation:
value = -75(years) + 250
================================
Cheers,
Stan H.
=================


Question 563928: Please help me solve this: Suppose that ln 2=a and ln 3=b use the properties of logarithms tro write each logarithm in terms of a and b.
ln27
Thanks

Answer by josmiceli(6777) About Me  (Show Source):

Question 563869: Help! Graph y=3e^x
Answer by Theo(2967) About Me  (Show Source):
You can put this solution on YOUR website!
graph%28600%2C600%2C-5%2C5%2C-10%2C500%2C3e%5Ex%29
it goes up very quickly.
assume e is equal to 2.718281828...
x is he exponent that e is raised to.
equation is y = 3*e^x
when x = -3, y = .15
when x = -2, y = .41
when x = -1, y = 1.1
when x = 0, y = 3
when x = 1, y = 8.15
when x = 2, y = 22.17
when x = 3, y = 60.26
when x = 4, y = 163.79
when x = 5, y = 445.24
the further positive x becomes, the further positive y = e^x becomes.
the further negative x becomes, the closer to 0 y becomes.
y = 3e^-20 would be equal to 6.18 * 10^-9.
y = 3e^-50 would be equal to 5.78 * 10^-22




Question 563705: Please help with this. I can't understand the way my teacher is trying to explain.
Let "b" be a number such that log(b)3 = 1.5 and log(b)5=2.2
evaluate the following:
1. log (b)15
2. log (b).6
3. log (b)125
4. log (b)(^3sqrt(3))^7
Thank you so much for your help

Found 2 solutions by Tatiana_Stebko, Alan3354:
Answer by Tatiana_Stebko(1060) About Me  (Show Source):
You can put this solution on YOUR website!
1. log+%28b%2C15%29=log+%28b%2C%283%2A5%29%29=log+%28b%2C3%29%2Blog+%28b%2C5%29=1.5%2B2.2=3.7
Use the properties of logarithm log+%28b%2C%28a%2Ac%29%29=log+%28b%2Ca%29%2Blog+%28b%2Cc%29
2.+log+%28b%2C0.6%29=log+%28b%2C%283%2F5%29%29=log+%28b%2C3%29-log+%28b%2C5%29=1.5-2.2=-0.7
Use the properties of logarithm log+%28b%2C%28a%2Fc%29%29=log+%28b%2Ca%29-log+%28b%2Cc%29
3. log+%28b%2C125%29=log+%28b%2C%285%5E3%29%29=3%2Alog+%28b%2C5%29=3%2A2.2=6.6
Use the properties of logarithm log+%28b%2C%28a%5En%29%29=n%2Alog+%28b%2Ca%29
4. log+%28b%2C+%28root%283%2C+3%29%29%5E7%29=log+%28b%2C+%28%283%29%5E%281%2F3%29%29%5E7%29=log+%28b%2C+%283%29%5E%287%2F3%29%29=%287%2F3%29%2Alog+%28b%2C+3%29=%287%2F3%29%2A1.5=3.5

Answer by Alan3354(21554) About Me  (Show Source):
You can put this solution on YOUR website!
Let "b" be a number such that log(b)3 = 1.5 and log(b)5=2.2
evaluate the following:
1. log (b)15
15 = 3*5
Multiplying --> adding the logs
log%28b%2C15%29+=+log%28b%2C3%29+%2B+log%28b%2C5%29+=+3.7
-----------------
2. log (b).6
0.6 = 3/5
Subtract logs when dividing
= log(3) - log(5) = -0.7
----------------
3. log (b)125
125+=+5%5E3
= 5*5*5
log(125) = log(5) + log(5) + log(5)
= 3*log(5)
Multiply the log by the exponent when raising to a power.
= 6.6
----------------
4. log (b)(^3sqrt(3))^7
If you mean log%28b%2C%28root%283%2C3%29%5E7%29%29
= 7%2Alog%28b%2C%28root%283%2C3%29%29%29 Multiply by the exponent
root%283%2C3%29+=+3%5E%281%2F3%29
log%283%5E%281%2F3%29%29+=+%281%2F3%29%2Alog%283%29+=+0.5 Multiply the log by the exponent
--> 7*0.5 = 3.5


Question 563670: Please help with this. I can't understand the way my teacher is trying to explain.
Let "b" be a number such that log(b)3 = 1.5 and log(b)5=2.2
evaluate the following:
1. log (b)5/3
2. log (b)3b^2
3. log (b)45
4. log (b)(b/15)
Thank you so much for your help

Answer by josmiceli(6777) About Me  (Show Source):
You can put this solution on YOUR website!
This is just making use of general rules of logs:
(1) +log%28a%2Ab%29+=+log%28a%29+%2B+log%28b%29+
(2) +log%28a%2Fb%29+=+log%28a%29++-+log%28b%29+
(3) +log%28a%5Eb%29+=+b%2Alog%28a%29+
(4) +log%28b%2Cb%29+=+1+
All these rules can use any base for the logs
-------------------------------------
+log%28+b%2C3+%29+=+1.5+
+log%28+b%2C5+%29+=+2.2+
-----------------
(1)
+log%28+b%2C5%2F3+%29+=+log%28+b%2C5+%29+-+log%28+b%2C3+%29+
+log%28+b%2C5%2F3+%29+=+2.2+-+1.5+
+log%28+b%2C5%2F3+%29+=+.7+
-----------------
(2)
+log%28+b%2C3b%5E2+%29+=+log%28b%2C3%29+%2B+2%2Alog%28b%2Cb%29+
+log%28+b%2C3b%5E2+%29+=+1.5+%2B+2%2A1+
+log%28+b%2C3b%5E2+%29+=+3.5+
-----------------
(3)
+log%28+b%2C45+%29+=+log%28b%2C9%29+%2B+log%28b%2C5%29+
+log%28+b%2C45+%29+=+log%28b%2C3%5E2%29+%2B+2.2+
+log%28+b%2C45+%29+=+2%2Alog%28b%2C3%29+%2B+2.2+
+log%28+b%2C45+%29+=+2%2A1.5+%2B+2.2+
+log%28+b%2C45+%29+=+5.2+
-----------------
(4)
+log%28+b%2Cb%2F15+%29+=+log%28+b%2Cb+%29+-+log%28+b%2C15+%29+
+log%28+b%2Cb%2F15+%29+=+1+-+log%28+b%2C3%2A5+%29+
+log%28+b%2Cb%2F15+%29+=+1+-+log%28+b%2C3+%29+-+log%28+b%2C5+%29+
+log%28+b%2Cb%2F15+%29+=+1+-+1.5+-+2.2+
+log%28+b%2Cb%2F15+%29+=+1+-+3.7+
+log%28+b%2Cb%2F15+%29+=+-2.7+


Question 563321: Find the value of the expression. Do not use decimals.
(1/3)log54+2log15+(1/3)log32-3log3
All the bases are 10

Answer by stanbon(48516) About Me  (Show Source):
You can put this solution on YOUR website!
(1/3)log54+2log15+(1/3)log32-3log3
------
log(2*27)^(1/3) + log(15^2) + log(8*4)^(1/3) - log(3^3)
------
= log(3*2^(1/3)) + log(225) + log(2*4^(1/3)) - log(27)
------
= log[3*2^(1/3)*225*2*4^(1/3)/27]
------
= log[1350*8^(1/3)/27]
------
= log[2700/27]
------
= log(100)
------
= 10
=====================
Cheers,
Stan H.
=====================


Question 562787: Please help me solve this:
2^-x=1.5
I tried this:
ln2^-x=ln1.5
-xln2=ln1.5
-x=ln1.5/ln2
x=-0.585
Thanks

Answer by issacodegard(60) About Me  (Show Source):
You can put this solution on YOUR website!
Yep, that's right!


Question 562635: solve for x:
500e^0.3x=600

Answer by issacodegard(60) About Me  (Show Source):
You can put this solution on YOUR website!
Hi, first divide by 500:
e%5E%280.3x%29=6%2F5
Take the natural logarithm of both sides:
ln%28e%5E%280.3x%29%29=ln%286%2F5%29
Simplify considering that ln(x) and e^x are inverse functions:
0.3x=ln%286%2F5%29
So we have,
x=ln%286%2F5%29%2F0.3


Question 562263: I'm having trouble solving this problem log 2x =-1. Can you give me an example of a problem close to this one?
Answer by issacodegard(60) About Me  (Show Source):
You can put this solution on YOUR website!
Hi, let say that log_b is the logarithm with base b function.
Solve log_2(3y+1)=6 for y
We have,
log_2(3y+1)=6
2^(log_2(3y+1))=2^(6) (exponentiate with respect to the base 2)
3y+1=64
3y=63
y=21
Note that log_b(x) and b^x are inverse functions, so b^(log_b(x))=x


Question 561805: PLEASE Help!
I have to answer these
An optical instrument is required to observe stars beyond the sixth magnitude, the limit of ordinary vision, however, even optical instruments have their limitations. The limiting magnitude L of an optical telescope with lens diameter D (in inches) is given by
L=8.8+5.1logD
a) Find the limiting magnitude for a homemade 6-inch reflecting telescope
Thank you so much!!

Answer by ankor@dixie-net.com(12684) About Me  (Show Source):
You can put this solution on YOUR website!
An optical instrument is required to observe stars beyond the sixth magnitude, the limit of ordinary vision, however, even optical instruments have their limitations. The limiting magnitude L of an optical telescope with lens diameter D (in inches) is given by
L = 8.8 + 5.1logD
:
a) Find the limiting magnitude for a homemade 6-inch reflecting telescope.
:
Just replace D with 6 and solve it
L = 8.8 + 5.1*log(6)
L = 8.8 + 5.1 * .778
L = 8.8 + 3.968857
L = 12.8 limiting magnitude


Question 561479: write 3^-3=1/27 as a logarithmic function
Answer by scott8148(5879) About Me  (Show Source):
You can put this solution on YOUR website!
log3(1/27) = -3


Question 560634: +9%5Ex%2B6%5Ex=2%5E%282x%2B1%29+
Found 2 solutions by richard1234, ankor@dixie-net.com:
Answer by richard1234(4789) About Me  (Show Source):
You can put this solution on YOUR website!
x = 0 works, but we need to prove it is the only solution. Note that we can write the equation as



Dividing both sides by 4^x,



As you can see, the LHS is a strictly increasing function of x (bijective or one-to-one), so there can only be one value of x in which the LHS equals 2. We have already found it to be x=0.

Answer by ankor@dixie-net.com(12684) About Me  (Show Source):
You can put this solution on YOUR website!
+9%5Ex%2B6%5Ex=2%5E%282x%2B1%29+
Just looking at this, you can see there is only one value for x that will make this work: x=0
+9%5E0%2B6%5E0=2%5E%282%280%29%2B1%29+
1 + 1 = 2^1


Question 560686: Hello, I have this problem with exponents, I know the answer but I don't know how to write this step by step:
8*3^100+3^100
Thank You !

Found 2 solutions by richard1234, jim_thompson5910:
Answer by richard1234(4789) About Me  (Show Source):
Answer by jim_thompson5910(21667) About Me  (Show Source):
You can put this solution on YOUR website!
8*3^100+3^100

(8+1)3^100

9*3^100


So 8*3^100+3^100 = 9*3^100
-------------------------------------------------------------------------------------------------
If you need more help, email me at jim_thompson5910@hotmail.com

Also, please consider visiting my website: http://www.freewebs.com/jimthompson5910/home.html and making a donation. Thank you

Jim

-------------------------------------------------------------------------------------------------




Question 560959: how do you solve 7^(x+4)=1?

Answer by stanbon(48516) About Me  (Show Source):
You can put this solution on YOUR website!
how do you solve 7^(x+4)=1?
---
Take the log of both sides:
(x+4)*log(7) = 0
---
x+4 = 0
---
x = -4
===============================
Cheers,
Stan H.
====================


Question 560650: Rewrite with rational exponents (√5mn)³
Answer by jim_thompson5910(21667) About Me  (Show Source):
You can put this solution on YOUR website!
%28sqrt%285mn%29%29%5E3+=+%28%285mn%29%5E%281%2F2%29%5E%27%27%29%5E3+=+%285mn%29%5E%283%2F2%29%5E%27%27


So %28sqrt%285mn%29%29%5E3+=+%285mn%29%5E%283%2F2%29%5E%27%27
-------------------------------------------------------------------------------------------------
If you need more help, email me at jim_thompson5910@hotmail.com

Also, please consider visiting my website: http://www.freewebs.com/jimthompson5910/home.html and making a donation. Thank you

Jim

-------------------------------------------------------------------------------------------------


Question 560620: 1) Solve the equation: x+1/11x=36
2) Solve the equation 5m+10=-6+10m
3)One number is twice another.If the largest number is m, how do you express the other number in terms of m?
4) Solve using the addition and multiplication principles , 1+2x<-20
5)Solve 3(4x-1)= 12

Answer by Maths68(1140) About Me  (Show Source):
You can put this solution on YOUR website!
1) Solve the equation: x+1/11x=36
x%2B1%2F11x=36
x%2B1=36%2811x%29
x%2B1=396x
or
396x=x%2B1
396x-x=1
395x=1
/ by 395 both sides
395x%2F395=1%2F395
cross%28395%29x%2Fcross%28395%29=1%2F395
x=1%2F395

2) Solve the equation 5m+10=-6+10m
5m%2B10=-6%2B10m
or
10m-6=5m%2B10
10m-5m=6%2B10
5m=16
/ by 5 both sides
5m%2F5=16%2F5
cross%285%29m%2Fcross%285%29=16%2F5
m=16%2F5
m=3%261%2F5
It is better to post single question in a post. For other questions post again.




Question 560393: +x%5E%282%28logx%29%5E2%29=10x%5E3+
Answer by Edwin McCravy(6931) About Me  (Show Source):
You can put this solution on YOUR website!
                  x2[log(x)]² = 10x3

Take logs of both sides:

             log{x2[log(x)]²} = log(10x3)

Use rules of logarithms:

        2[log(x)]²·log(x) = log(10) + log(x³)

               2[log(x)]³ = 1 + 3·log(x)

2[log(x)]³ - 3·log(x) - 1 = 0

let u = log(x)

             2u³ - 3u - 1 = 0

Possible rational solutions for u are ±1, ±1%2F2

Try 1:

1|2  0 -3 -1
 |   2  2 -1
  2  2 -1 -2 

No that isn't a solution, since it did not give a remainder of 0.

Try -1:

-1|2  0 -3 -1
  |  -2  2  1
   2 -2 -1  0 

Yes -1 is a solution, so we have factored

             2u³ - 2u - 1 = 0
as

     (u + 1)(2u² - 2u - 1) = 0

   u + 1 = 0    2u² - 2u - 1 = 0
       u = -1             u = %28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
                          u = %28-%28-2%29+%2B-+sqrt%28%28-2%29%5E2-4%2A%282%29%2A%28-1%29+%29%29%2F%282%2A%282%29%29+
                          u = %282+%2B-+sqrt%284%2B8+%29%29%2F4+
                          u = %282+%2B-+sqrt%2812%29%29%2F4+
                          u = %282+%2B-+2sqrt%283%29%29%2F4+
                          u = %282%281+%2B-+sqrt%283%29%29%29%2F4+
                          u = %281+%2B-+sqrt%283%29%29%2F2+
                          
The log equation u = log(x) is equivalent to the exponential
equation x = 10u

So we have three solutions:

x = 10-1, x = +matrix%282%2C1%2C%27%27%2C10%5E%28%28%281%2Bsqrt%283%29%29%2F2%29%29%29%29, x = +matrix%282%2C1%2C%27%27%2C10%5E%28%28%281-sqrt%283%29%29%2F2%29%29%29%29

In decimal approximations they are

x = 0.1,  x = 23.22872667,  x = 0.4305014278

Edwin


Question 560336: 3logx^2-(log(-x))^2=9
Answer by stanbon(48516) About Me  (Show Source):
You can put this solution on YOUR website!
3logx^2-(log(-x))^2 = 9
-------------------
6log(x) + [log(x)]^2 = 9
----
(log(x))^2 + 6log(x) - 9 = 0
Factor:
[log(x)-3]^2 = 0
---
log(x)-3 = 0
---
log(x) = 3
---
x = 10^3
----
x = 1000
================
Cheers,
Stan H.
================


Question 559518: If +log%28+a%2C+x+%29+=+P+ and +log%28+a%2C+y+%29+=+7+ . Express +log%28+a%2C+x%5E2y+%29+ in terms of P and 7.
+log%28+a%2C+x+%29+=+P+ => +a%5EP+=+x+
+log%28+a%2C+y+%29+=+7+ => +a%5E7+=+7+
Log(a,(a^P)^2a^7 ) => Log( a, a^2Pa^7 )
and I was stuck. I am not even sure of what i'm doing ????
So, please help.

Found 2 solutions by Theo, htmentor:
Answer by Theo(2967) About Me  (Show Source):
You can put this solution on YOUR website!
your problem states:
log(a,x) = p
log(a,y) = 7
you want to express log(a,x^2*y) in terms of p and 7.
start with log(a,x^2*y)
this becomes:
log(a,x^2) + log(a,y) which becomes:
2*log(a,x) + log(a,y)
since log(a,x) = p and log(a,y) = 7, your equation becomes:
2*p + 7
it can also be shown as:
2p + 7
the logarithm rules you are using are:
log(x*y) = log(x) + log(y)
log(x^y) = y*log(x)
these rules apply regardless of the base of the logarithm that is being used.

Answer by htmentor(580) About Me  (Show Source):
You can put this solution on YOUR website!
If +log%28+a%2C+x+%29+=+P+ and +log%28+a%2C+y+%29+=+7+ . Express +log%28+a%2C+x%5E2y+%29+ in terms of P and 7.
===================================
For simplicity, we write log_a as log
p = log(x)
7 = log(y)
Using the rules of logarithms, we can write
log(x^2*y) = log(x^2) + log(y) = 2*log(x) + log(y) = 2p + 7


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