Problem
For $\log _{5} 20$, (a) Estimate the value of the logarithm between two consecutive integers. For example, $\log _{2} 7$ is between 2 and 3 becaus $2^{2}<7<2^{3}$. (b) Use the change-of-base formula and a calculator to approximate the logarithm to 4 decimal places. (c) Check the result by using the related exponential form. Part: $0 / 3$ Part 1 of 3 (a) Estimate the value of the logarithm between two consecutive integers. \[ \square<\log _{5} 20<\square \]
Solution
Step 1 ：Estimate the value of the logarithm between two consecutive integers. We need to find two consecutive integers such that \(5^{n}<20<5^{n+1}\).
Step 2 ：Start by checking the powers of 5. We know that \(5^1 = 5\) and \(5^2 = 25\).
Step 3 ：So, \(5^1<20<5^2\). Therefore, \(\log _{5} 20\) is between 1 and 2.
Step 4 ：Final Answer: \(\boxed{1<\log _{5} 20<2}\)
