Question

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!

Answer 1

duplicate

Answer 2

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 right-hand-side,
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)

Answer 3

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 right-hand-side,
log x = 1/2 (log a + log b - log c)
= (1/2)log a + (1/2)log b <b>-</b> (1/2)log c
= log(√a) + log(√b) <b>-</b> log(√c)
= log(√a √b <b>/</b> √c)
= log(√(ab<b>/</b>c))

Assuming log() stands for logarithm to the base e,
e^{log x} = e^{log(√(ab<b>/</b>c))}
x = √(ab<b>/</b>c)