Logarithms question (1 Viewer)

Drongoski · Aug 13, 2015

rand_althor said:
$$If $2^x=5^y=10^z$, using logarithms to base 10, or otherwise, show that $\frac{1}{z}=\frac{1}{x}+\frac{1}{y}$.$

Let log be the common log, i.e. to the base 10

$log 2^x = log 5^y = log 10^z = z \\ \\ \therefore xlog 2 = y log 5 = z \\ \\ \therefore xlog2 = z \implies \frac {1}{x} = \frac {log 2}{z}\\ \\ $and $ ylog 5 = z \implies \frac {1}{y} = \frac {log 5}{z} \\ \\ \therefore \frac {1}{x} + \frac {1}{y} = \frac {log 2 + log 5}{z} = \frac {log (2 \times 5)}{z} = \frac {1}{z}$

rand_althor · Aug 13, 2015

Thanks for the help!

Logarithms question (1 Viewer)

rand_althor

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Drongoski

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rand_althor

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