Question

write the following logarithm in terms of the natural logarithm. then use a calculator to find the value of the logarithm, rounding your result to four decimal places. \\(\log_{5} 20\\) write the logarithm in terms of the natural logarithm. \\(\log_{5} 20 = \square\\)

Explanation:

Step1: Apply Change of Base Formula

The change of base formula for logarithms is $\log_b a = \frac{\ln a}{\ln b}$ (or also $\frac{\log a}{\log b}$). For $\log_5 20$, we can use the natural logarithm (ln) version. So we have $\log_5 20=\frac{\ln 20}{\ln 5}$.

Step2: Calculate Numerator and Denominator

First, calculate $\ln 20$. Using a calculator, $\ln 20\approx 2.995732274$. Then calculate $\ln 5$, which is approximately $1.609437912$.

Step3: Divide and Round

Now divide the two results: $\frac{2.995732274}{1.609437912}\approx 1.8607$. Round this to four decimal places.

Answer:

$\log_5 20\approx 1.8607$ (and in terms of natural logarithms, $\log_5 20 = \frac{\ln 20}{\ln 5}$)