MathLOGS
posted by Jess on .
Express x in terms of a,b and c.
log x = 1/2 (log a + log b  log c)
Please solve and explain how to do this type of problem, thank you!

duplicate

To solve
log x = 1/2 (log a + log b  log c)
we need to know some properties of logarithms, namely:
1. log(a)+log(b) = log(ab)
2. (1/2)log(a)=log(a^{1/2})=log(√(a))
3. e^{ln x } = x, or
10^{log10 x} = x
Proceeding to simplify the righthandside,
log x = 1/2 (log a + log b  log c)
= (1/2)log a + (1/2)log b + (1/2)log c
= log(√a) + log(√b) + log(√c)
= log(√a √b √c)
= log(√(abc))
Assuming log() stands for logarithm to the base e,
e^{log x} = e^{log(√(abc))}
x = √(abc) 
2nd rule of logarithm should read:
2. (1/2)log(a)=log(a^{1/2})=log(√(a))
and the solution has to be corrected because of an erroneous sign:
Proceeding to simplify the righthandside,
log x = 1/2 (log a + log b  log c)
= (1/2)log a + (1/2)log b  (1/2)log c
= log(√a) + log(√b)  log(√c)
= log(√a √b / √c)
= log(√(ab/c))
Assuming log() stands for logarithm to the base e,
e^{log x} = e^{log(√(ab/c))}
x = √(ab/c)