To the nearest thousandth, log_102 is 0.301 and log_103 is 0.477. Which of the following is the best approximation of log_5 10?

1979 AMC 12 Problem 18

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18.

To the nearest thousandth,

\log_{10}2

0.301

and

\log_{10}3

0.477.

Which of the following is the best approximation of

\log_5 10?

\frac87

\frac97

\frac{10}{7}

\frac{11}{7}

\frac{12}{7}

Use

\log_{10}5=1-\log_{10}2

Apply change of base, then compare the decimal result with the five fractions

We have

\log_{10}5\approx1-0.301=0.699.

Therefore

\begin{aligned} \log_5 10&=\frac1{\log_{10}5}\\ &\approx\frac1{0.699}\\ &\approx1.431. \end{aligned}

Of the choices,

\frac{10}{7}\approx1.429

is closest.

Therefore, the correct answer is C.