What is the solution to this logarithm equation?

[nvez]

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.

 

[viciousp]

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

 

[Dick]

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.

 

[nvez]

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.