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

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Question 752465: solve for x for the equation:
X^2(10^x)-x10^x=2(10^x)
Thank you :)
Answer by MathTherapy(1430)

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solve for x for the equation:
X^2(10^x)-x10^x=2(10^x)
Thank you :)

x%5E2%2810%5Ex%29+-+x%2810%5Ex%29+=+2%2810%5Ex%29

x%5E2%2810%5Ex%29+-+x%2810%5Ex%29+-+2%2810%5Ex%29+=+0

10%5Ex%2A%28x%5E2+-+x+-+2%29+=+10%5Ex%2A%280%29

------ Factoring out GCF, 10%5Ex

(x - 2)(x + 1) = 0

highlight_green%28x+=+2%29

You can do the check!!

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Question 751956: Which of the following is true about the function below?
(1)/(√(x+4))
Answer by tommyt3rd(617)

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please repost to include the options :)

Question 751815: The population of a colony of rabbits grows exponentially. The colony started with 10 rabbits and five years later, there were 340 rabbits.
1) Write a formula for the population of the colony of rabbits as a function of the number of years since it was founded (but not "e" base).
2) How many rabbits will there be TEN years after the start of the colony?
3) Approximately how long will it take for the population of the colony to reach 1000 rabbits?

Thank you in advance for your time and assistance.
Answer by josgarithmetic(1540)

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Look at each year symbolically step by step.
Let r = the growth rate in a year in fractional form, or decimal form.
1 year-------- 10%2A%281%2Br%29

2 year -------- 10%2A%281%2Br%29%5E2

3 year -------- 10%2A%281%2Br%29%5E3

4 year ------- 10%2A%281%2Br%29%5E4

5 year ------- highlight%2810%2A%281%2Br%29%5E5=340%29

The function will be p%28x%29=10%281%2Br%29%5Ex

where x is the number of years. You would use the highlighted '340' equation to find the growth rate, r and then use this in p(x) accordingly.

Question 751743: how would I solve log(5x+7)=2??

thanks!
Answer by tommyt3rd(617)

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%0D%0A10%5Elog%285x%2B7%29=10%5E2%0D%0A

Question 751125: the max population that earth can sustain is 40 billion. if the current population is 4.2 billion, in how many years will the max population be reached? Assume that each year the population is 2% more than the previous year.
Found 2 solutions by timvanswearingen, nerdybill:
Answer by timvanswearingen(106)   (Show Source):
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Start with 4,200,000,000 and multiply by 1.02. Continue to multiply each product by 1.02 until you reach 40,000,000,000 and you'll have your answer. The number of times you multiplied is the number of years it will take.

Answer by nerdybill(6963)   (Show Source):
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the max population that earth can sustain is 40 billion. if the current population is 4.2 billion, in how many years will the max population be reached? Assume that each year the population is 2% more than the previous year.
.
Exponential growth equation:
A = Pe^(rt)
the problem give us:
P is 4.2
A is 40
r is .02
.
40 = 4.2e^(.02t)
40/4.2 = e^(.02t)
ln(40/4.2) = .02t
ln(40/4.2)/.02 = t
112.69 = t
or
113 years = t

Question 750949: subtract and simplify
4/(x+1) - (2-x)/x
Answer by tommyt3rd(617)   (Show Source):
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subtract and simplify

:)

Question 750903: You invest $1,300 in an account with an annual interest rate of 2.5%, compounded monthly. How much money is in the account after 5 years? Round your answer to the nearest whole number.
Answer by stanbon(57412)   (Show Source):
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(1/2)ln(x-3)-ln(x+2)
----
= ln(x-3)^(1/2) - ln(x+2)
----
= ln[(x-3)^(1/2)/(x+2)]
==========================
cheers,
Stan H.
===========

Question 750798: 4^2x+1 = 671

Thank you for your help!
Answer by timvanswearingen(106)   (Show Source):
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Subtract 1 from both sides:

Take log base 4 of both sides. All you are left with on the left side is 2x. So 2x= log base 4 of 670. Using a change of base formula, 2x= (log 670) / (log 4). Divide both sides by 2, x is about 2.347.

Question 750564: Write an exponential function whose graph passes through the points (0, -6) and (-2, -54)
Answer by MathLover1(6646)   (Show Source):
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Solved by pluggable solver: Find the equation of line going through points

hahaWe are trying to find equation of form y=ax+b, where a is slope, and b is intercept, which passes through points (x1, y1) = (0, -6) and (x2, y2) = (-2, -54).
Slope a is .
Intercept is found from equation , or . From that,
intercept b is , or .

y=(24)x + (-6)

Your graph:

since you can't see second point on graph above, here is another graph:

Question 750427: How do you solve this problem: -3ex<-7?
I tried to figure it out. I could only get this far > -3ex=-7ex
Answer by MathLover1(6646)   (Show Source):
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...switch sides to get positive values

or

..........

Question 750242: please help me solve this problem it is a logarithmic, I think. As a town gets smaller, the population of its high school decreases by 12% each year. the student body numbers 125 students now. in how many years will it number about 75 students?
Answer by josgarithmetic(1540)   (Show Source):
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Simplest way to analyze it is that each year population is pop*(1.00-0.12).

Starting with 125 students,
1 year 125*(1-0.12)
2 year
3 year
n year

You are looking for how
then

Can you take this the rest of the way to the solution for n? Take logarithms of both sides, and use the rule about logs,

Question 750171: 3.5 x 7^(4x-1)= 1000^(3-x)
find what x equals
Answer by Alan3354(30993)   (Show Source):
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3.5 x 7^(4x-1)= 1000^(3-x)
log(3.5) + (4x-1)*log(7) = (3-x)*log(1000) = (3-x)*3
log(7) - log(2) + (4x-1)*log(7) = 9 - 3x
log(7) - log(2) + 4x*log(7) - log(7) = 9 - 3x
4xlog(7) + 3x = 9 + log(2)
x*(4log(7) + 3) = log(2) + 9
x = (log(2) + 9)/(4log(7) + 3)
----------
x = (log(2) + log(10^9))/(log2401 + log(1000))
x = log(2E9)/log(2401000)
x =~ 1.457752088

Question 749560: I can not seem to figure this question out...
log[4](x+3)-log[4](x+2) >= 3/2
Answer by KMST(1874)   (Show Source):
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For and to exist, it must be that <-->
(because logarithm exist only for positive numbers).
--> --> --> --> --> -->
Since , multiplying both sides times does not require flipping the inequality sign, so
--> --> --> --> -->
The solution is

Question 749425: Explain how to write 8a^9/125a^6 as an expression with only one exponent
Answer by checkley79(3056)   (Show Source):
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8a^9/125a^6
8a^3/125 ans.

Question 749134: Logx-log2=3
Answer by nerdybill(6963)   (Show Source):
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logx-log2=3
log(x/2)=3
x/2 = 10^3
x/2 = 1000
x = 2000

Question 749112: how do I solve ln (5-x) =12 ?
Answer by timvanswearingen(106)   (Show Source):
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Natural Log has base e, so you'll raise e to both sides of the equation:

The e and ln basically cancel each other out, so you've got:

Now just solve for x:

Add x to both sides:

Subtract e^12 from both sides:

That is an exact answer. If you want an approximated answer, just evaluate that and round to whichever place you need to.

Question 748885: 3^(2x-1)=6^(x+2)
Answer by lwsshak3(6523)   (Show Source):
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3^(2x-1)=6^(x+2)
(2x-1) log(3)=(x+2)log(6)
(2x-1)/(x+2)=log(6)/log(3)≈1.6309
2x-1=1.6309(x+2)
2x-1=1.6309x+3.2619
0.3691x=4.2619
x≈11.5467

Question 748655: These questions confuse me a bit and this site has helped me before. Thank you, please show me how to solve these problems (for x) .
16) log(x+4)-log6=1
18) log(x-1)^2=log(-5x-1)
please help. thank you again so much.
Found 2 solutions by MathTherapy, stanbon:
Answer by MathTherapy(1430)   (Show Source):
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please show me how to solve these problems (for x) .
16) log(x+4)-log6=1
18) log(x-1)^2=log(-5x-1)
please help. thank you again so much.

16) log(x+4)-log6=1

x + 4 = 60 ------- Cross-multiplying

x = 60 - 4, or

18) log(x-1)^2=log(-5x-1)

(x + 1)(x + 2) = 0

, or

You can do the check on both!!

Send comments and “thank-yous” to “D” at MathMadEzy@aol.com

Answer by stanbon(57412)   (Show Source):
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log(x+4)-log6=1
log[(x+4)/6] = 1
(x+4)/6 = 10^1
x+4 = 60
x = 56
----------------------------
18) log(x-1)^2=log(-5x-1)
(x-1)^2 = -5x-1
x^2 -2x + 1 = -5x -1
-----
x^2 +3x = 0
x(x+3) = 0
x = 0 or x = -3
------------------
Cheers,
Stan H.

Question 748647: log(x+9)=log(2x-7)
I do not need the answer so much as how to do this.
Answer by timvanswearingen(106)   (Show Source):
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You raise 10 to the power of both sides...the 10 and log cancel out, so you end up with just x+9=2x-7 and you can solve from there.

Question 748638: Hello, any help to resolve this problem will be appreciated.
The data points shown came from an experiment that involved tungsten (W) and iridium (Ir). Write the equation that expresses tungsten as a function of iridium: W = mIr + b
Tungsten in grams (vertical): 20 40 60 80 100
Iridium in grams (horizontal): 2 4 6 8 10
If you need more information just let me know.
Thanks
Answer by Alan3354(30993)   (Show Source):
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The data points shown came from an experiment that involved tungsten (W) and iridium (Ir). Write the equation that expresses tungsten as a function of iridium: W = mIr + b
Tungsten in grams (vertical): 20 40 60 80 100
Iridium in grams (horizontal): 2 4 6 8 10
----------------------
W = 10Ir

Question 748384: I keep getting a no soulution for 16^n < 8^(n+1) but in the back of the book it says its N<3
Answer by JoeTaxpayer(145)   (Show Source):
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16^n < 8^(n+1)
(2^4)^n < (2^3)^n+1
2^4n < 2^(3n+3)
4n < 3n+3
n < 3

Question 748288:
Found 2 solutions by tommyt3rd, Alan3354:
Answer by tommyt3rd(617)   (Show Source):
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we will denote log base 5 as log5

:)

Answer by Alan3354(30993)   (Show Source):
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-----------
Sub u for 5^(x-1)
u = 5u - 5
u = 5/4
-------
5^(x-1) = 5/4
5^x = 25/4
x*log(5) = log(25/4)
x = log(6.25)/log(5)
x =~ 1.13864688

Question 747891: Can I please get help to find the distance between (1,9) and (5,10) and if necessary round the answer to two decimal places.
And also if I can get help to find the midpoint of the line segment whose points are (3,2) and (7,8). Thanks
Answer by Alan3354(30993)   (Show Source):
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find the distance between (1,9) and (5,10) and if necessary round the answer to two decimal places.

d =~ 4.12
=======================
And also if I can get help to find the midpoint of the line segment whose points are (3,2) and (7,8)
It's the average of x & y separately.
For x: (3+7)/2 = 5
For y: (2+8)/2 = 5
--> (5,5)

Question 747752: differentiate and simplify y = 9ln(5/x)
Answer by Alan3354(30993)   (Show Source):
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differentiate and simplify y = 9ln(5/x)
-----------
y = 9ln(5) - 9ln(x)
y' = -9/x

Question 747347: At the beginning of an experiment, a scientist has 244 grams of radioactive goo. After 240 minutes, her sample has decayed to 3.8125 grams. what is the half life of goo in minutes? Find a formula for G(t), the amount of goo remaining at time t. How many grams of goo will remain after 47 minutes.
Answer by stanbon(57412)   (Show Source):
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At the beginning of an experiment, a scientist has 244 grams of radioactive goo. After 240 minutes, her sample has decayed to 3.8125 grams. what is the half life of goo in minutes?
----
A(t) = Ao*(1/2)^(t)
------
3.8125 = 244*(1/2)^(240/k)
-----
0.0157 = (1/2)^(240/k)
---
(240/k) = log(0.0157)/log(0.5) = 5.9953
---
k = 240/5.9953
----
k = 6 minutes
---------
Find a formula for G(t), the amount of goo remaining at time t. How many grams of goo will remain after 47 minutes.
---
A(47) = 244*(1/2)^(47/6)
---
A(47) = 244*(1/2)^7.833
A(47) = 1.0698 grams
=========================
Cheers,
Stan H.
===================

Question 747279: Solve 1 n 2+ 1 n x=5
Answer by nerdybill(6963)   (Show Source):
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I'm assuming you meant:
ln(2) + ln(x) =5
ln(2x) =5
2x = e^5
x = (e^5)/2
x = 74.21

Question 747285: In the equation y=ab^(x-h)+k, how does the value of k affect the graph?
Answer by nerdybill(6963)   (Show Source):
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In the equation y=ab^(x-h)+k, how does the value of k affect the graph?
.
The 'k' is the "vertical shift".
If positive, moves graph up k units.
If negative, moves graph down k units.

Question 747071: please help me solve
log base 6 X + log base 6 X-3=2
the answer is 7.365 but I need to know how to solve. thank you.
Answer by lwsshak3(6523)   (Show Source):
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log base 6 X + log base 6 X-3=2
the answer is 7.365
***

convert to exponential form: base(6) raised to log of number(2)=number((x)(x-3))
6^2=x^2-3x
x^2-3x-36=0
solve for x by quadratic formula:
x≈-4.68 (reject, x>0)
or
x≈7.684

Question 747012: how do you do log7(2-x)=log7(5x)
Answer by tommyt3rd(617)   (Show Source):
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we use the inverse property of logs/exponents:

and the inputs are recovered:

Question 746949: Solve for x: (e^x)+(e^x)= 0
Found 2 solutions by Alan3354, jim_thompson5910:
Answer by Alan3354(30993)   (Show Source):
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Solve for x: (e^x)+(e^x)= 0
-------------
2e^x = 0
e^x = 0
No solution.

Answer by jim_thompson5910(28598)   (Show Source):
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(e^x)+(e^x)= 0

2e^x= 0

e^x= 0/2

e^x= 0

ln(e^x) = ln(0)

Since you cannot take the natural log of 0, there are no solutions.

Question 746904: find the equation of the tangent line to the graph of
a. f(x) = -2x^2+3x-2, at x=-2
Answer by Cromlix(353)   (Show Source):
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First obtain your y coordinate by
substituting x = -2 into f(x)
f(x) = -2(-2)^2 + 3(-2) - 2
= -8 -6 -2
= -16 Coords are: {-2, -16}
Differentiate f(x)
f'(x) = - 4x + 3
Substitute x = -2 into f'(x)
-4(-2) + 3 = 11 (gradient)
Using y - b = m(x - a)
y + 16 = 11(x + 2)
y = 11x + 22 - 16
y = 11x + 6

Question 746193: Please help me solve this equation. I am having trouble finding all the steps.
log(16+2x)=log(x^2-4x)
Found 2 solutions by savvyhush23, josgarithmetic:
Answer by savvyhush23(50)   (Show Source):
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Please help me solve this equation. I am having trouble finding all the steps.
log(16+2x)=log(x^2-4x)
.
.
Is it finding x value in order for the function to be equal?
.
.
If that so, since both side is in logarithm, you can eliminate log leaving:
move 16 + 2x to the right,

the function is a quadratic equation, therefore to find x, use quadratic formula:
a = 1, b = -6 and c = -16

.
.
Therefore the answer are 8 and -2

Answer by josgarithmetic(1540)   (Show Source):
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You read that as, in a more general way, the logarithm of this expression equals the logarithm of that expression. So we must expect that "this" expression is equal to "that" expression. There is a way to make this much more formal, but doing so should be unnecessary.

No steps to get confused, unless you are unfamiliar with quadratic equations.

Question 746192: Please help me solve this equation: 4log(3)-4log(8)
Answer by Alan3354(30993)   (Show Source):
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Please help me solve this equation: 4log(3)-4log(8)
------------
Solve for what? There's no variable.
And, it's not an equation, there's no equal sign.

Question 746048: A to the power of -1 is equal to 1 please prove me how
Answer by rothauserc(210)   (Show Source):
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Please note that
A^-1 = 1/A
and
1/A = 1 if A=1

Question 745760: log(A)=log(A[0])+0.1tlog(.8) Solve for A as a function of t.
Having a bit of difficulty on where to start here. Please help an old man going back to school...
Answer by KMST(1874)   (Show Source):
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I would start with properties of logarithms, like
and

EXTRA:
as a function of is an exponential decay function.
The graph looks like this

If you start with a quantity and lose 20% of what you have every 10 hours, at time hours you will have 80% of left

At time hours you will have 80% of that 80% left
, which is 64% of
At time hours you will have 80% of 80% of 80% left
, which is 51.2% of

Question 745685: X^3/4= 8 solve the missing base
Answer by josgarithmetic(1540)   (Show Source):
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X is raised to a certain power and gives 8. Try reversing both sides. Raise both sides to the (4/3) power and see how this works. BOTH sides.

An extremely simple EXAMPLE for comparison is this:
.

If you have , then undo this by doing .

Question 745655: The question was: Sketch the graph of the function f(x)=3^(x+1). Indicate the y-intercept and the horizontal asymptote. Can anyone help? Please?
Answer by josgarithmetic(1540)   (Show Source):
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Think what happens as x gets more and more negative. x+1 will become more and more negative. will be , which approaches zero, because in this form, the exponent is positive and increases without bound. Horizontal asymptote is y=0.

Question 745491: 16^(x+4)=8^(3x-2)
Answer by tommyt3rd(617)   (Show Source):
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For this example our strategy is to make the bases equal, then we can set the exponents equal:

so x=22/5

Question 745058: e raised to (x squared +6) equals e raised to 5x. Solve for x
Answer by nerdybill(6963)   (Show Source):
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e raised to (x squared +6) equals e raised to 5x. Solve for x
.

take ln of both sides:

x = {2,3}

Question 744527: Hi, just hoping that I put this question in the right category. I have a table, and I just can't seem to make an exponential equation out of it. I have tried many times, only to have it work for a couple of the x values. I tried ones such as 2^x+2 - 1 and many more. Thanks in advance for any help given! Here's my table
x f(x)
0 4
1 7
2 13
3 25
4 49
5 97
6 193
Answer by lynnlo(4164)   (Show Source):
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