# Math Help - Logarithmic identities problem - solves one way but not another

1. ## Logarithmic identities problem - solves one way but not another

Hi,

I was working through a log problem using logarithmic identities. The problem is worded as follows:

log(4) = 0.703x
log(1/4) = z

solve for z in terms of x

To solve this, I tried one approach, which is simply:
z = log(1/4) = log(4^-1) = -1*log(4) = -0.703x.

This seems to be the correct answer. Then I also tried solving the same question by adding the two equations together:

log(4) + log(1/4) = 0.703x + z

so: z = log(4*(1/4)) - 0.703x = -0.703x. The solution in this case is also in agreement.

My question is... why does it not work when you subtract one equation from the other? My work comes out like this:

log(4) - log(1/4) =0.703x - z
log[4/[1/4]] = .703x - z
log(16) = .703x - z
z = .703x - log(16)

The z in this case doesn't agree with z in the other cases. Can anyone tell me what I am doing wrong in the last case? Thanks.

2. You've not done anything wrong, just didn't finish it off.

$\log(16) = \log(4^2) = 2\log(4)$ and since you have $\log(4) = 0.703x$ then your equation becomes $z = 0.703x - 2 \cdot 0.703x = -0.703x$

3. Thank you so much! Had me confused for a while...