Question

use natural logarithms to evaluate the logarithm to the nearest thousandth.\\(\log _{5}10\\)\\(\log _{5}10 = \square\\) (round to the nearest thousandth as needed.)

Explanation:

Step1: Recall Change of Base Formula

The change of base formula for logarithms is $\log_{b}a = \frac{\ln a}{\ln b}$ (or $\frac{\log a}{\log b}$). Here, $b = 5$ and $a = 10$, so we use natural logarithms (ln) to rewrite $\log_{5}10$ as $\frac{\ln 10}{\ln 5}$.

Step2: Calculate Natural Logarithms

First, find $\ln 10$ and $\ln 5$. We know that $\ln 10 \approx 2.302585093$ and $\ln 5 \approx 1.609437912$.

Step3: Divide the Values

Now, divide $\ln 10$ by $\ln 5$: $\frac{2.302585093}{1.609437912} \approx 1.420$.

Answer:

1.420