# Thread: conversion of log to the base of 4 to log to the base of 2

1. ## conversion of log to the base of 4 to log to the base of 2

i know that log to the base of a of b can be re-written as (log of b)/(log of a)
But, i got lost when i saw a question
log to the base of 16 (x) + log to the base of 4 (x) + log to the base of 2 (x) = 7

how would you go about solving this question?
I thought about re-defining every single term of those logs in to (log of b)/(log of a) shape, but it doesnt give me a whole number answer (answer is 16)

2. If you use LaTex it is a lot easier to read.
$$\log_{16}(x)$$ gives $\displaystyle \log_{16}(x)$.
$$\frac{\log_{16}(x)}{\log_2(x)}$$ gives $\displaystyle \frac{\log_{16}(x)}{\log_2(x)}$.

3. Thanks for the info, i will rewrite it to make it more clear
$\displaystyle \log_{16}(x)$ + $\displaystyle \log_{4}(x)$+ $\displaystyle \log_{2}(x)$ = 7
i couldnt find x (the hint in the book says i should be able to change $\displaystyle \log_{16}(x)$ and $\displaystyle \log_{4}(x)$ to the $\displaystyle \log_{2}(x)$ form
but i cant do that.
how should i make that happen?

4. $\displaystyle \log_{16}(x)=\frac{\log(x)}{\log(16)}=\frac{\log(x )}{4\log(2)}=\frac{1}{4}\log_2(x)$.

5. thank you so much.