Questions on Algebra: Logarithm answered by real tutors!

Tutors Answer Your Questions about logarithm (FREE)

Question 571127: 4.135* 27.5 * 8.529
Answer by rfer(10417)

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=969.8539

Question 571026: solve for x: log(base 4) x^9=6
Answer by nyc_function(2626)

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Use the definition of logarithms to write in exponential form.

So, log(base 4) x^9 = 6 becomes 4^6 = x^9.

We now solve for x.

4^6 = 4096

4096 = x^9

Take ninth root of both sides of the equation.
The answer for x is 2(9throot{8}) or in decimal form x is approximately
2.5198 and that's it.

Question 570935: how would you solve a problem solving for x: ln (x/630) = 1.2317?
Answer by Alan3354(21541)

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how would you solve a problem solving for x: ln (x/630) = 1.2317?
-----
x%2F630+=+e%5E1.2317

Question 570820: Alice can vacuum in 40 minutes. It takes Bob 45 minutes to do the same job. Find the time it takes Alice and Bob to do the job together
Answer by nerdybill(5399)

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Alice can vacuum in 40 minutes. It takes Bob 45 minutes to do the same job. Find the time it takes Alice and Bob to do the job together
Let x = time (minutes) it takes for both
then
x(1/40 + 1/45) = 1
multiplying (40)(45):
x(45 + 40) = (40)(45)
x(45 + 40) = 1800
x(85) = 1800
x = 1800/85
x = 21.18 minutes

Question 570808: how do u solve?: 3.5x^-4 = 175
please show work i don't understand how to do it.
Answer by KMST(576)

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3.5x%5E-4+=+175

Applying logarithms to both sides we get
log%28%283.5x%5E-4%29%29+=log%28%28+175%29%29

Logarithm of a product is the sum of the logarithms, so we can write that as
log%28%283.5%29%29%2Blog%28%28x%5E-4%29%29+=log%28%28+175%29%29

Logarithm of a power is the exponent multiplied by the logarithm of the base, so we can write that as
log%28%283.5%29%29%2B%28-4%29log%28%28x%29%29+=log%28%28+175%29%29

log%28%283.5%29%29%2B%28-4%29log%28%28x%29%29+=log%28%28+175%29%29

It's plain algebra from now on. Subtracting log%28%283.5%29%29%7D%7D+from+both+sides%2C+we+get%0D%0A%7B%7B%7B%28-4%29log%28%28x%29%29+=log%28%28+175%29%29-log%28%283.5%29%29

log%28%283.5%29%29%7D%7D+from+both+sides%2C+we+get%0D%0A%7B%7B%7B%28-4%29log%28%28x%29%29+=log%28%28+175%29%29-log%28%283.5%29%29

Dividing both sides by (-4) we get
log%28%28x%29%29+=%28log%28%28+175%29%29-log%28%283.5%29%29%29%2F%28-4%29

log%28%28x%29%29+=%28-1%2F4%29%28log%28%28+175%29%29-log%28%283.5%29%29%29

That would be enough to get an approximate answer for the value of

If we want an exact answer we can apply some more logarithm properties.
We know that log%28%28a%2Fb%29%29=log%28%28a%29%29-log%28%28b%29%29

, so we can write the equation above as
log%28%28x%29%29+=%28-1%2F4%29%28log%28%28+175%2F3.5%29%29%29

and since

log%28%28x%29%29+=%28-1%2F4%29%28log%28%28+50%29%29%29

Because logarithm of a power is the exponent multiplied by the logarithm of the base, so we can write that as
log%28%28x%29%29+=log%28%2850%5E%28-1%2F4%29%29%29

log%28%28x%29%29+=log%28%2850%5E%28-1%2F4%29%29%29

For a more elegant way to express the value of x,
1%2Froot%284%2C50%29=%281%2Froot%284%2C50%29%29%28root%284%2C50%29%5E3%2Froot%284%2C50%29%5E3%29=root%284%2C50%29%5E3%2Froot%284%2C50%29%5E4=root%284%2C50%29%5E3%2F50=root%284%2C50%5E3%29%2F50=root%284%2C125000%29%2F50=root%284%2C625%2A200%29%2F50=root%284%2C625%29%2Aroot%284%2C200%29%2F50=5%2Aroot%284%2C200%29%2F50=root%284%2C200%29%2F10

1%2Froot%284%2C50%29=%281%2Froot%284%2C50%29%29%28root%284%2C50%29%5E3%2Froot%284%2C50%29%5E3%29=root%284%2C50%29%5E3%2Froot%284%2C50%29%5E4=root%284%2C50%29%5E3%2F50=root%284%2C50%5E3%29%2F50=root%284%2C125000%29%2F50=root%284%2C625%2A200%29%2F50=root%284%2C625%29%2Aroot%284%2C200%29%2F50=5%2Aroot%284%2C200%29%2F50=root%284%2C200%29%2F10

Question 570809: please solve help me solve this word problem: Lucy bought a car for $25,000. The value of the car depreciates by about 13% each year. when will the value of the car be $10,000 ?
please help me. and please show me the work to get the answer because i have no idea how to solve this problem.
Answer by josmiceli(6766)

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Suppose for now the car starts out being worth

Let the yearly depreciation = +a+

( .13, actually )
At end of 1st yr it is worth
+x+-+a%2Ax+

At end of 2nd yr
+x+-+a%2Ax+-+a%2A%28+x+-+a%2Ax+%29+

At end of 3rd yr
+x%2A%28+1+-+a+%29%5E3+

------------------------
You can see the progression here
In your problem, +x+=+25000+

and

I can say

where

is the number of yrs car is owned
+%28+1+-+.13+%29%5En+=+10000+%2F+25000+

+n+=+%28+-.39794+%29+%2F+%28+-.06048+%29+

and

So, after 6 yrs ( 6 yrs and 7 months ),
the car is worth $10,000
---------------------------
I can check this
+25000%2A%28+1+-+.13+%29%5E6.579+=+10000+

close enough
The key to understanding this problem is to see
that each years depreciation is on last year's
DEPRECIATED value. Hope this helps.

Question 570411: log base 5 (sqrt(125))=
Answer by nerdybill(5399)

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log base 5 (sqrt(125))
log base 5 (sqrt(5^3))
log base 5 ((5^3)^(1//2))
log base 5 (5^(3/2))
3/2

Question 570340: the expression log8 64 is equivalent to?
8 2 1/2 1/8
Answer by Alan3354(21541)

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8%5E2+=+64

Question 570185: okay i need to condense into a single logarithm
2logx+3log7-6logz
Answer by josmiceli(6766)

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The general rules are +log%28a%2Ab%29+=+log%28a%29+%2B+log%28b%29+

and

+log%28a%2Fb%29+=+log%28a%29+-+log%28b%29+

Just use the rules in reverse
also, +b%2Alog%28a%29+=+log%28a%5Eb%29+

-------------------------
+2%2Alog%28x%29+%2B+3%2Alog%287%29+-+6%2Alog%28z%29+

+log%28x%5E2%29+%2B+log%287%5E3%29+-+log%28z%5E6%29+

Question 569126: Given log(base a)5 = 2.32, and log(base a)9=3.17, find log(base a) 25/9= amd find log(base a) radical 45 =
Answer by lwsshak3(2900)

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Given log(base a)5 = 2.32, and log(base a)9=3.17, find log(base a) 25/9= amd find log(base a) radical 45 =
**
log(base a)=loga
loga(25/9)=loga(5^2/9)=2loga(5)-loga(9)=2*2.32-3.17=1.47
loga(√45)=loga(45^.5)=.5loga(45)=.5loga(5*9)=.5(loga(5)+loga(9))=.5(2.32+3.17)=2.745

Question 569232: e^(2x+1)-10=62
Answer by lwsshak3(2900)

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e^(2x+1)-10=62
e^(2x+1)=72
take log of both sides
(2x+1)lne=ln72
lne=1
2x+1=ln72
2x=ln72-1
x=(ln72-1)/2
using calculator
x≈1.6383

Question 569367: How do you solve for x in the equation 7.2^x-4=8.21?

Answer by lwsshak3(2900)

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How do you solve for x in the equation
7.2^x-4=8.21?
7.2^x=8.21+4=12.21
take log of both sides
xlog7.2=log12.21
x=log12.21/log7.2
using calculator
x≈1.2676
..
Check:
7.2^1.2676-4≈8.21

Question 569885: What is the value of v, v=log base4 1024
Answer by lwsshak3(2900)

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What is the value of v, v=log base4 1024
Exponental form: base(4) raised to log of number(v)=number(1024)
4^v=1024
4^5=1024
v=5

Question 569943: Log a-2 log b +3 log c (Rewrite the expression as a single log)
Answer by lwsshak3(2900)

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Log a-2 log b +3 log c (Rewrite the expression as a single log)
=log[ac^3/b^2]

Question 570001: For all x>2, log(x-2)+ log x = ?
A) log (-2)
B) log(2x-2)
C) log (x^2-2x)
D) log(x-2/x)
E) log( x/x-2)
Answer by lwsshak3(2900)

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For all x>2, log(x-2)+ log x = ?
Place under a single log using multiplication rule for logs
log(x-2)+ log x=log[x(x-2)]
=log(x^2-2x)
C is the correct answer.

Question 570110: find the value of log 0.001
Answer by nerdybill(5399)

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log 0.001
log 10^(-3)
-3

Question 570095: log question
11-6^x=3
gabe1001@comcast.net

Answer by nerdybill(5399)

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11-6^x=3
11 = 3+6^x
8 = 6^x
take log base 6 of both sides:
log(8)/log(6) = x
1.161 = x

Question 570037: Yes, hello I am doing Logarithms and I cannot solve these two problems:
log(x+3)=log8-log2
ln sqrt(e)
Please any help is appreciated
Answer by Alan3354(21541)

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log(x+3)=log8-log2
log(x+3) = log(4)
x+3 = 4
x = 1
----------------
ln sqrt(e)
= ln%28e%5E%281%2F2%29%29

= 1/2

Question 570002: For all x>2, log(x-2)+ log x = ?
A) log (-2)
B) log(2x-2)
C) log (x^2-2x)
D) log(x-2/x)
E) log( x/x-2)
Answer by jim_thompson5910(21667)

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Hint: use the idea that log(x)+log(y) = log(xy)

Question 569984: Expand the expression log 4X^5
Answer by jim_thompson5910(21667)

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log(4x^5) = log(4)+log(x^5) = log(4)+5log(x)

So log(4x^5) = log(4)+5log(x)

Question 569464: 123=500e^-0.12x
Answer by ankor@dixie-net.com(12678)

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Assume the problem is:
123=500e%5E%28-0.12x%29

rewrite it
500e%5E%28-0.12x%29

= 123
divide both sides by 500
e%5E%28-0.12x%29

= .246
:
Using nat logs
ln%28e%5E%28-0.12x%29%29

= ln(.246)
:
the log equiv of exponents
-.12x*ln(e) = ln(.246)
;
find e%5Ex

of both sides, the nat log of e is 1
-.12x = -1.4024
;
x = %28-1.4024%29%2F%28-.12%29

x = +11.6869
:
:
We can check this on a good calc: enter 500*e^(-.12*11.6869). results 122.999 ~ 123

Question 569318: 3+(2b)=11 b=? how do i do this
Answer by solver91311(12114)

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Add -2b to both sides.

Divide both sides by 9.

Reduce the fraction.

John

My calculator said it, I believe it, that settles it

The Out Campaign: Scarlet Letter of Atheism

Question 569083: Write the expression below as a single logarithm. Please show all of your work.
ln(x^3+8)-ln(x-4)-ln(x+2)
Answer by KMST(576)

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ln%28x%5E3%2B8%29-ln%28x-4%29-ln%28x%2B2%29=ln%28x%5E3%2B8%29%2B%28-1%29ln%28x-4%29%2B%28-1%29ln%28x%2B2%29=+ln%28%28x%5E3%2B8%29%28x-4%29%5E-1%28x%2B2%29%5E-1%29=ln%28%28x%5E3%2B8%29%281%2F%28x-4%29%29%281%2F%28x%2B2%29%29%29=ln%28%28x%5E3%2B8%29%2F%28%28x-4%29%28x%2B2%29%29%29=ln%28%28%28x%2B2%29%28x%5E2-2x%2B4%29%29%2F%28%28x-4%29%28x%2B2%29%29%29

ln%28x%5E3%2B8%29-ln%28x-4%29-ln%28x%2B2%29=ln%28x%5E3%2B8%29%2B%28-1%29ln%28x-4%29%2B%28-1%29ln%28x%2B2%29=+ln%28%28x%5E3%2B8%29%28x-4%29%5E-1%28x%2B2%29%5E-1%29=ln%28%28x%5E3%2B8%29%281%2F%28x-4%29%29%281%2F%28x%2B2%29%29%29=ln%28%28x%5E3%2B8%29%2F%28%28x-4%29%28x%2B2%29%29%29=ln%28%28%28x%2B2%29%28x%5E2-2x%2B4%29%29%2F%28%28x-4%29%28x%2B2%29%29%29

=
=

highlight%28ln%28%28x%5E2-2x%2B4%29%2F%28x-4%29%29%29

Question 569119: I need help on how to show my work when solving this problem: e^.05t = 1600
Answer by Alan3354(21541)

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e%5E.05t+=+1600

Take the log of both sides
0.05t = ln(1600) =~ 7.37776
t =~ 147.555

Question 569080: Solve the logarithmic equation. Please show all of your work.
2log4(x+3)=2
Answer by josmiceli(6766)

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+2%2Alog%28+x+%2B+3+%29+=+2+

Make substitution +1+=+log%2810%29+

check:

Question 568846: Please help me solve this problem:
2log(x)-log(7)=log(252)
Solve for x
Answer by 10301650(25)

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x=42

Question 568329: 10000e^(-0.06t) = 1000
Please help me solve for t :D
Answer by scott8148(5869)

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dividing by 10000 ___ e^(-0.06t) = 1/10

rules for exponents ___ e^(.06t) = 10

taking natural log ___ .06t = ln(10)

dividing by .06 ___ t = [ln(10)] / .06

Question 566988: What is the exponential form of (square root of 7)^5
Answer by jim_thompson5910(21667)

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%28sqrt%287%29%29%5E5+=+%287%5E%281%2F2%29%5E%27%27%29%5E5+=+7%5E%285%2F2%29%5E%27%27

%28sqrt%287%29%29%5E5+=+7%5E%285%2F2%29%5E%27%27

Question 567717: solve the following equation for x: 27^(5-x) = 9^(x-2)
Answer by jim_thompson5910(21667)

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27^(5-x) = 9^(x-2)

(3^3)^(5-x) = (3^2)^(x-2)

3^(3(5-x)) = 3^(2(x-2))

Since the bases are equal, the exponents are equal.

3(5-x)=2(x-2)

15-3x=2x-4

-3x=2x-4-15

-3x-2x=-4-15

-5x=-4-15

-5x=-19

x=(-19)/(-5)

x=19/5

Question 567879: I am trying to solve
e^.05t = 1600
I have tried and just can't figure it out
Answer by jim_thompson5910(21667)

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e^.05t = 1600

ln(e^.05t) = ln(1600)

.05t*ln(e) = ln(1600)

.05t*1 = ln(1600)

.05t = ln(1600)

t = ln(1600)/(0.05)

t = 147.555178164558

Note: the last answer is approximate.

Question 567309: Express 2log 3 + 3log 2 - log 6 as a single logarithm.
Answer by Alan3354(21541)

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Express 2log 3 + 3log 2 - log 6 as a single logarithm.
-----
2log(3) = log(9)
3log(2) = log(8)
----
--> log(8*9/6)
= log(12)

Question 566887: 1.51^x=70. How do i solve for x?
Found 2 solutions by richard1234, solver91311:
Answer by richard1234(4787)   (Show Source):
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If your calculator can evaluate log of any base, then take the log base 1.51 of both sides to get

If not, then use the other tutor's solution.

Answer by solver91311(12114)   (Show Source):
You can put this solution on YOUR website!

Take the log of both sides -- any base, it doesn't matter.

Use

to write:

Then

The rest is calculator work.

John

My calculator said it, I believe it, that settles it

Question 567199: how do you solve 3.5^x=47.9 without using a calculator? I get really confused when it comes to the decimals..
Answer by stanbon(48502)   (Show Source):
You can put this solution on YOUR website!
3.5^x=47.9
----
You have no choice but to take the log of both sides:
x(log(3.5)) = log(47.9)
----
x = log(47.9)/(log(3.5))
----
x = 3.0885
------
Unless you use a slide rule of log tables you cannot
get this answer without a calculator.
----
Cheers,
Stan H.
=============

Question 566804: 3 = log down 5 x + log down 5 (x+20) soe help
Answer by solver91311(12114)   (Show Source):
You can put this solution on YOUR website!

What does "soe help" mean?

John

My calculator said it, I believe it, that settles it

Question 566479: Solve the following system for (x,y):
1. log9(x)+ logy(8)= 2
2. logx(9)+ log8(y)= (8/3)
The term attached to log is the base, the term in parenthesis the argument. This feels like it should be simple, but after 30 minutes of useless substituting, I figure I'm missing an obvious connection . . .
Thanks to anyone who sees it.
TS
Answer by KMST(576)   (Show Source):
You can put this solution on YOUR website!
I'll connect you.
Let's change variables.
and
The first equation transforms easily:
-->
For the next one, we have to use a property of logarithms.
As with all the properties of logarithms, you can remember it, or you can rediscover it from the definition of logarithm every time you need it.
I always opted for the second choice, but in this case, rediscovery is not that easy, so I say understand the proof once, and then try to remember that

To help students remember that property, a popular tutor used to call it "skewer it and turn it upside down." That helped because he would write a very large above the and pretend it was a toothpick skewering an olive, before pretending to turn it upside down and take it to his mouth, and
the he would then write on the board sort of looks like an upside down . So
, so and
, so
With that, the second equation gets transformed
--> -->
So we have two equations
and
and just need to find w and z. No logarithm worries (for now).
At this point I would divide the first equation by the second equation
= --> -->
If you figure out here that the numbers and must be and , good for you.
I had to remember that they would be solutions of the quadratic equation
<--> and use the quadratic formula.
It could be that and ,
or that and .
The first solution:
and
--> -->
and --> -->
The second solution:
and
--> -->
--> -->

Question 566246: expand the logarithm log base 2 12^5/11^2
Answer by Alan3354(21541)   (Show Source):
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expand the logarithm log base 2 12^5/11^2
--------
= 5log(12) - 2log(11) any base

Question 565992: Topic : Differentiation
(Q) The line (l) is the tangent, at the point (2,3), to the curve with equation y = a + bx^2, where a and b are constants. The tangent (l) has gradient 8. Find the values of a and b ?
Answer by htmentor(579)   (Show Source):
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(Q) The line (l) is the tangent, at the point (2,3), to the curve with equation y = a + bx^2, where a and b are constants. The tangent (l) has gradient 8. Find the values of a and b ?
============================
The slope (gradient) of the tangent line to the curve y = a + bx^2 is
dy/dx = 2bx = 2b(2) = 4b
Since the gradient is 8, we have the value for b:
4b = 8 -> b = 2
Now use the equation for the curve to solve for a:
(x,y) = (2,3)
3 = a + 2*2^2
3 = a + 8
This gives a = -5
So the curve is y = -5 + 2x^2

Question 565998: Topic : Surds
Rationalise the denominator and simplify the following surd

3sqrt5 + 2
-------------
2sqrt5 - 4

Answer by ad_alta(170)   (Show Source):
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1) (3*sqrt(5)+2)/(2*sqrt(5)-4) [multiply the top and bottom by 2*sqrt(5)+4]
2) (16*sqrt(5)+38)/(4)
3) (8*sqrt(5)+19)/2

Question 565979: Solve the simultaneous equations :

logbase2(x) + logbase8(y) = -1
logbase4(x) + logbase2(y) = 2

Cannot solve, spent hours on it
Any help would be appreciated
Answer by lwsshak3(2900)   (Show Source):
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Solve the simultaneous equations :
logbase2(x) + logbase8(y) = -1
logbase4(x) + logbase2(y) = 2
**
Change to base2:
log2(x) + log2(y)/log2(8) = -1
log2(x)/log2(4) + log2(y) = 2
..
log2(4)=2
log2(8)=3
..
log2(x) + log2(y)/3= -1
log2(x)/2 + log2(y) = 2
..
3log2(x) + log2(y)= -3
log2(x) +2 log2(y) =4
..
6log2(x) + 2log2(y)= -6
log2(x) +2 log2(y) =4
subtract
6log2(x)-log2(x)=-10
log2(x^6)-log2(x)=-10
log2(x^6/x)=-10
log2(x^5)=-10
convert to exponential form: base(2) raised to log of number(-10)=number(x^5)
2^-10=x^5
take 5th root of both sides
x=(2^-2)=1/(2^2)=1/4
..
solving for y
log2(x) +2 log2(y) =4
log2(2^-2)+2log2(y)=4
-2+2log2(y)=4
2log2(y)=6
log2(y)=3
y=8
ans:
x=1/4
y=8

Question 565756: what is the answer to { { { log(z-3)=2? } } }
Do you bring the 3 out first?
Answer by unlockmath(1118)   (Show Source):
You can put this solution on YOUR website!
Hello,
Let's write this as an exponential:
10^2=z-3
Rewritten as:
100=z-3
Add 3 tho both sides to get:
z=103
Make sense?
RJ
www.math-unlock.com

Question 565317: How do I solve:
Log x + log (13 -3x) = 1
(Bases are 10)
Answer by scott8148(5869)   (Show Source):
You can put this solution on YOUR website!
adding logs means multiplying their arguments ___ and ___ 1 = log(10)

x(13 - 3x) = 10 ___ 13x - 3x^2 = 10 ___ 0 = 3x^2 - 13x + 10

factoring ___ 0 = (3x - 10)(x - 1)

Question 565397: Every £1 of money invested in a savings scheme continuously gains interest at a rate of 4% per year. Hence, after x years, the total value of an initial£1 investment is £y, where y = 1.04^x.
(a) Calculate, to the nearest £, the total value of an initial £800 investment after 10 years ?
(b) Use logarithms to find the number of years it takes to double the total value of any initial investment ?
Answer by stanbon(48502)   (Show Source):
You can put this solution on YOUR website!
Every £1 of money invested in a savings scheme continuously gains interest at a rate of 4% per year. Hence, after x years, the total value of an initial£1 investment is £y, where y = 1.04^x.
------------------
(a) Calculate, to the nearest £, the total value of an initial £800 investment after 10 years ?
A(10) = 800*1.04^10 = $1184.20
-------------------------------------
(b) Use logarithms to find the number of years it takes to double the total value of any initial investment ?
--
Solve: 2 = 1*1.04^x
1.04^x = 2
----
x(log(1.04) = log(2)
--
x = log(2)/log(1.04)
x = 17.67 years
======================
Cheers,
Stan H.
==============

Question 565009: solve the equation for x express answer in deciamal value log16^8-log25^5=x
Answer by lwsshak3(2900)   (Show Source):
You can put this solution on YOUR website!
solve the equation for x express answer in deciamal value
log16^8-log25^5=x
log(16^8/25^5)=x
convert to exponential form: base(10) raised to log of number(x) = number(16^8/25^5)
10^x=(16^8/25^5)≈439.80465..
take log of both sides
xlog10=log439.80465..
x≈2.6433