Problem 32 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, use a calculator to evaluate the logarithm. Round to three decimal places. $$ \log _{6} 10 $$

Step-by-Step Solution

Answer

After applying steps as mentioned above, the computed value of \(\log_{6}10\) to three decimal places will be the result.

Step 1: Understanding Logarithms

In this exercise, there is a function \(\log_{6}10\). This is a logarithmic function, specifically a logarithm with base 6. It asks, 'To what power must we raise 6 to get a value of 10?'

Step 2: Evaluating the Logarithm

The next step is to evaluate the logarithm. That can be done by applying the change of base formula, which says, the \(\log_{b}a = \frac{\log a}{\log b}\) where \(\log\) here is the natural logarithm. Following this formula, \(\log_{6} 10 = \frac{\log 10}{\log 6}\). To proceed, we require a calculator for this.

Step 3: Result and Rounding

After performing the calculation with a calculator, we get a decimal value. It's specified in the problem to round this value to three decimal places, which results in a particular value.

Key Concepts

Understanding Logarithmic Functions

Logarithmic functions are inverse functions to exponential ones. They answer the question: 'To what power must we raise the base to get a certain number?' A logarithmic function is usually written as \( \text{log}_b(x) \) where \( b \) is the base and \( x \) is the number you want to find the exponent for.

For example, in our exercise, we are working with the function \( \text{log}_6(10) \), which denotes the power to which 6 must be raised to equal 10. Understanding this function is crucial in various fields such as science, engineering, and finance, where growth and decay patterns are analyzed, and where they help solve for time or rates in compound interest formulas.

Applying the Change of Base Formula

The 'Change of Base Formula' is a method used to evaluate logarithms on a calculator that may only have keys for base 10 or the natural logarithm (base \( e \)). This formula states that \( \text{log}_b(a) = \frac{\text{log}_k(a)}{\text{log}_k(b)} \), with \( k \) being any positive number.

In practical terms, this allows us to calculate logarithms with bases that are not directly supported by our calculators. To solve the exercise, we use \( \text{log}_6(10) = \frac{\text{log}(10)}{\text{log}(6)} \) where \( \text{log} \) could represent either base 10 or the natural logarithm, as both are commonly available functions on calculators.

The Natural Logarithm

The natural logarithm, denoted by \( \text{ln} \), is a logarithm with base \( e \), where \( e \) is an irrational and transcendental number approximately equal to 2.71828. It has unique properties that make it especially useful in mathematics, particularly in calculus due to its relationship with the exponential function \( e^x \).

The natural logarithm is pivotal in solving problems involving growth and decay, such as populations or radioactive decay, and in complex calculations like integration and differentiation where the function \( e^x \) naturally appears due to its derivative being itself.

Solving Exponential Equations

Exponential equations feature an unknown variable in the exponent, such as \( b^x = a \) and are often solved using logarithms. To isolate the variable, one can take the logarithm of both sides, allowing for the exponent to be brought down front as a coefficient, thanks to logarithmic properties. This simplifies the equation to \( x \text{log}(b) = \text{log}(a) \), which can then be solved for \( x \).

This method can be applied to a broad range of problems, including compound interest calculations and modeling exponential growth or decay. Logarithms thus serve as a powerful tool for unravelling the complexities inherent in exponential functions.