Write the following logarithms in terms of the natural logarithm. Then use a calculator to find the value of the logarithm, rounding your result to four decimal places.


${\log }_{6}60$

Verified step by step guidance

1. Step 1: Understand the problem. We need to express \( \log_6 60 \) in terms of natural logarithms.
2. Step 2: Use the change of base formula for logarithms, which states that \( \log_b a = \frac{\ln a}{\ln b} \).
3. Step 3: Apply the change of base formula to \( \log_6 60 \), giving us \( \log_6 60 = \frac{\ln 60}{\ln 6} \).
4. Step 4: Use a calculator to find \( \ln 60 \) and \( \ln 6 \).
5. Step 5: Divide the result of \( \ln 60 \) by \( \ln 6 \) to find the value of \( \log_6 60 \), rounding to four decimal places.

Key Concepts

Change of Base Formula

The Change of Base Formula allows us to convert logarithms from one base to another. Specifically, for any logarithm \\( ext{log}_b(a) \\), it can be expressed as \\( rac{ ext{log}_k(a)}{ ext{log}_k(b)} \\) for any positive base \\( k \\. This is particularly useful for converting logarithms to the natural logarithm, \\( ext{ln} \\, (base \\ e) \\, which is commonly available on calculators.

Natural Logarithm

The natural logarithm, denoted as \\( ext{ln}(x) \\, is the logarithm to the base \\ e \\, where \\ e \\, is approximately equal to 2.71828. It is a fundamental concept in calculus and is used extensively in various applications, including growth models and integration. Understanding how to manipulate natural logarithms is essential for solving logarithmic equations.

Calculator Usage for Logarithms

Using a calculator to evaluate logarithms involves inputting the correct values and understanding the functions available. Most scientific calculators have dedicated buttons for natural logarithms and may require the use of the Change of Base Formula for other bases. Familiarity with the calculator's functions is crucial for accurately computing logarithmic values and rounding them to the desired precision.