What is the solution to this logarithm equation?

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In summary, the first steps seem to be more correct but when you distribute that 2 you forgot to multiply by lnx by 2.
  • #1
nvez
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Homework Statement


Find x in the following equation

(ln 64 / 2) + ln x = ln 18 - ln x

The answer should be x = 3/2

Homework Equations


Logarithm laws

ln m - ln n = ln (m/n)

The Attempt at a Solution


I tried two methods:

(ln 64 / 2) + ln x = ln 18 - ln x
ln 32 + ln x = ln 18 - ln x
ln 32 - ln 18 = - ln x - ln x
ln (32/18) = -2 ln x
ln (32/18) = ln x-2
(32/18) = x-2
(32/18)1/-2 = x
0.75 = 3/4 = x

Wrong answer

(ln 64 / 2) + ln x = ln 18 - ln x
ln 64 + ln x = 2 (ln 18 - ln x)
ln 64 + ln x = 2 ln 18 - 2 ln x
ln 64 + ln x = ln 182 - ln x2
ln 64 - ln 182 = - ln x - ln x2
ln 64 - ln 324 = - ln x - 2 ln x
ln 64 - ln 324 = - 3 ln x
ln 64 - ln 324 = ln x-3
ln (64/324) = ln x-3
(64/324) = x-3
(64/324)1/-3 = x
1.717 = x

Wrong answer again.

The first steps seem to be more right to me though but I can't find what I did incorrect..

Thank you in advanced. :smile:
 
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  • #2
You've got the right idea but you are either messing your rules or not distributing properly.

For the first one you did (ln64)/2 is eqaul to ln(32) which is not true

The second one is more correct but when you distribute that 2 you forgot to multiply by lnx by 2.

Your second line should look like this:
ln 64 + 2(ln x) = 2 (ln 18 - ln x)

you should be able to solve it from there
 
  • #3
I think you are interpreting the first term wrong. Instead of ln(64/2) do it with ln(64)/2. That will give you x=3/2.
 
  • #4
Thank you, I have resolved my problem with the following:

ln 64 + 2 (ln x) = 2 (ln 18 - ln x)
ln 64 + 2 ln x = 2 ln 18 - 2 ln x
ln 64 - 2 ln 18 = -2 ln x - 2 ln x
ln 64 - 2 ln 18 = -4 ln x
ln 64 - ln 324 = ln x-4
ln (64/324) = ln x-4
(64/324) = x-4
(64/324)1/-4 = x
1.5 = x

Thanks again, this is such a helpful resource.
 

FAQ: What is the solution to this logarithm equation?

What is a logarithm equation?

A logarithm equation is an equation in which the variable appears in the argument of a logarithm function. It involves finding the value of the variable that makes the equation true.

How do I solve a logarithm equation?

To solve a logarithm equation, you can use the property of logarithms that states: logb(x) = y if and only if by = x. This means that you can rewrite the equation as an exponential equation and solve for the variable. Remember to check for extraneous solutions.

What are the common bases used in logarithm equations?

The most common bases used in logarithm equations are base 10 (common logarithm) and base e (natural logarithm). However, any positive number can be used as a base for logarithm equations.

Can logarithm equations have multiple solutions?

Yes, logarithm equations can have multiple solutions. This is because logarithms are not one-to-one functions, meaning that different inputs can result in the same output. It is important to check for extraneous solutions when solving logarithm equations.

What are some real-life applications of logarithm equations?

Logarithm equations have many real-life applications, such as in finance for calculating compound interest, in biology for measuring the intensity of earthquakes and the pH scale, and in computer science for data compression and signal processing.

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