Problem 17 - Precalculus

Question

Evaluate the expression. \(\log _{6}(216)\)

Step-by-Step Solution

Answer

\(\log _{6}(216) = 3\)

Step 1: Understand the Expression

The expression \(\log _{6}(216)\) is asking you to find the power to which the base 6 must be raised to obtain 216.

Step 2: Change of Base Formula

To find \(\log _{6}(216)\), we can use the change of base formula which states that \[ \log_{b}(a) = \frac{\log_{10}(a)}{\log_{10}(b)} \] or using natural logarithms as \[ \log_{b}(a) = \frac{\ln(a)}{\ln(b)} \] depending on calculator availability.

Step 3: Numerically Evaluate

Using a calculator, evaluate \( \log_{6}(216) \) as \[ \frac{\log_{10}(216)}{\log_{10}(6)} = \frac{2.334}{0.778} \approx 3 \] or using natural logarithms \[ \frac{\ln(216)}{\ln(6)} \approx 3 \]. Therefore, \(\log _{6}(216) = 3\).

Key Concepts

Change of Base Formula

The change of base formula is a helpful way to simplify logarithmic expressions, especially when you need to compute them using a standard calculator. Often, calculators only have functions for logarithms base 10 (common logarithms) and base e (natural logarithms).

To overcome this, we can use the change of base formula:

• For common logarithms: \( \log_{b}(a) = \frac{\log_{10}(a)}{\log_{10}(b)} \)
• For natural logarithms: \( \log_{b}(a) = \frac{\ln(a)}{\ln(b)} \)

This formula allows you to convert any logarithm with a different base into a form that your calculator understands. In essence,

• compute the logarithm of the number in the numerator, and
• compute the logarithm of the base in the denominator.

By dividing these results, you translate an otherwise complex evaluation into simple arithmetic.

Logarithmic Expressions

Logarithmic expressions are mathematical statements that use the concept of logarithms. A logarithm answers the question: "To what power must a base be raised to obtain a certain number?"

• In the expression \( \log_{b}(x) \), "\( x \)" is the number you wish to express as a power of "\( b \)",
• "\( b \)" is the base,
• and the result indicates the power to which \( b \) must be raised to yield \( x \).

For example, in the expression \( \log_{6}(216) \), we are seeking a power that 6 must be raised to in order to get 216.

Understanding how to interpret logarithmic expressions is essential in unraveling the question they pose. Through practice, reading and evaluating these expressions will become second nature.

Numerical Evaluation

Numerical evaluation of logarithmic expressions involves using arithmetic and possibly a calculator to find a numerical approximation. When the numbers involved do not yield an obvious or simple answer, calculators come handy, especially with the change of base formula.

Let's break it down using the example \( \log_{6}(216) \). By applying the change of base formula, we evaluate the expression by performing the following calculations:

• First, calculate \( \log_{10}(216) \) or \( \ln(216) \) using a calculator.
• Then, calculate \( \log_{10}(6) \) or \( \ln(6) \) similarly.
• Finally, divide the first result by the second. This will give the power to which 6 must be raised to obtain 216.

The calculated value should be approximately 3, indicating that 6 cubed equals 216. This method simplifies finding solutions to logarithms that don't yield easily calculated answers.