# Logarithm Equation Problem

1. Jan 18, 2009

### nvez

1. The problem statement, all variables and given/known data
Find x in the following equation

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

The answer should be x = 3/2

2. Relevant equations
Logarithm laws

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

3. 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

(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

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

2. Jan 18, 2009

### 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

3. Jan 18, 2009

### 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.

4. Jan 18, 2009

### 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.