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August 28, 2014

August 28, 2014

Posted by **Jess** on Monday, December 21, 2009 at 4:32pm.

log x = 1/2 (log a + log b - log c)

Please solve and explain how to do this type of problem, thank you!

- Math-LOGS -
**Anonymous**, Monday, December 21, 2009 at 4:41pmduplicate

- Math-LOGS -
**MathMate**, Monday, December 21, 2009 at 4:53pmTo 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)

- correction -
**MathMate**, Tuesday, December 22, 2009 at 11:59am2nd 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**-**(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)

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