Problem 41 - College Algebra Essentials

Question

Evaluate each expression without using a calculator. $$8^{\log _{8} 19}$$

Step-by-Step Solution

Answer

The value of the expression \(8^{\log _{8} 19}\) is \(19\).

Step 1: Identify the Main Logarithmic Rule

The main rule used in this exercise is \(a^{\log_a(b)} = b\). This is one of the basic rules of logarithms, stating that if a certain number \(a\) is raised to a logarithm of the same base \(a\), then the results equals the number within the logarithm.

Step 2: Apply the Logarithmic Rule

Applying this rule to our expression \(8^{\log _{8} 19}\), the base of the exponent and the base of the logarithm are the same, being \(8\). Hence, we can directly apply the logarithmic rule to simplify this expression, making it equivalent to the number inside the logarithm, which is \(19\).