# Express log value from other log values

#### Ilikebugs

I got a really messy answer (2+(3/(1+a/(2-a))))/(a+2b), is there an answer that looks clearer?

I think I'll come back to this later. I started with how I would approach it.

log(28)/log(35) = log(4*7)/log(5*7) = [log(4) + log(7)] / [log(5) + log(7)] = [2 log(2)/log(7) + 1] / [log(5)/log(7) + 1]

a = log(49) / log(14) = 2 log(7) / log(2*7) = 2 log(7) / [log(2) + log(7)] = 2 / [log(2)/log(7) + 1]. Thus log(2)/log(7) = 2/a - 1
b = log(5) / log(14) = log(5) / log(7*2) = log(5) / [log(7) + log(2)] = 1 / [log(7)/log(5) + log(2)/log(5)]. Thus log(5)/log(7) = 1/b - log(2)/log(5)

log(2)/log(5) = log(2)/log(7) * log(7)/log(5) = (2/a - 1) * (1 / (1/b - log(2)/log(5))
From here, isolate log(2)/log(5). (breakpoint)

topsquark

#### Idea

View attachment 39629I got a really messy answer (2+(3/(1+a/(2-a))))/(a+2b), is there an answer that looks clearer?
yes, there is an answer that looks simpler

$$\displaystyle \frac{4-a}{a+2 b}$$

topsquark
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}