Problem 87 - College Algebra

Question

Describe the power rule for logarithms and give an example.

Step-by-Step Solution

Answer

The power rule for logarithms states that the exponent within the logarithm can be treated as a coefficient: \( log_b \, (a^x) = x \cdot log_b \, a \). An instance of this rule: \( log_2 (8^3) = 3 \cdot log_2 8 \), which eventually simplifies to 9.

Step 1: Defining Power Rule for Logarithms

The power rule is a basic theorem in logarithms that states that for any number \(x\) and any positive integer \(a\), \(log_b \, (a^x) = x \cdot log_b \, a\). This rule is very useful when simplifying logarithmic expressions.

Step 2: Creating and solving an example

Let's apply the power rule of logarithms to the following example: Calculate \( log_2 (8^3) \).Using the power rule, the expression can be rewritten as: \( log_2 (8^3) = 3 \cdot log_2 8 \). Since \( 2^3 = 8 \), \( log_2 8 = 3 \). So,\( 3 \cdot log_2 8 = 3 \cdot 3 = 9 \).

Step 3: Interpreting Results

From the solution, it can be inferred that the power rule is a straightforward way to manage powers inside a logarithm. The presented solution also shows that applying this rule, the expression simplifies to a simple multiplication.