Problem 22 - Introductory and Intermediate Algebra for College Students 4th

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 log notation

In this problem, we're dealing with logarithms. We have \(\log_{7} 49\). This is read as 'log base 7 of 49'. This notation represents the exponent or power to which the base number (7) must be raised to obtain the number inside the log (49).

Step 2: Transform the log equation into an equivalent exponential equation

Use the formula for converting a logarithmic equation into an exponential equation. The logarithmic equation \(\log_{b} n = x\) is equivalent to the exponential equation \(b^x = n\). Therefore, for \(\log_{7} 49\), the equivalent exponential form is \(7^x = 49\)

Step 3: Solve the equation for x

To finish solving, a knowledge of the powers of 7 is needed. We know that \(7^2 = 49\), thus we take \(x=2\) . Therefore, x, which is the value of \(\log_{7} 49\), equals to 2.