Problem 22 - College Algebra Essentials

Question

Evaluate each expression without using a calculator. $$\log _{7} 49$$

Step-by-Step Solution

Answer

The value of \(\log_{7} 49\) is 2.

Step 1: Understand the problem

The expression to evaluate is \(\log_{7} 49\). This is a logarithm to the base 7, and the value inside the logarithm is 49. Therefore, the problem is asking 'To what power do you need to raise 7 to get 49?'.

Step 2: Apply the definition of a logarithm

Using the definition of a logarithm: if \(b^y = x\), then \(\log_b x = y\). Here, the base \(b\) is 7 and the number \(x\) is 49. Our task is to find the exponent \(y\). Since \(7^2 = 49\), it's clear that \(y\) should be 2.