Problem 19 - Algebra 2

Question

Evaluate each logarithm. $$ \log _{49} 7 $$

Step-by-Step Solution

Answer

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

Step 1: Identify the base and number

Here, the base of the logarithm is 49 and the number is 7.

Step 2: Express the base as power of number

Express 49 as a power of 7. It is equal to \(7^2\).

Step 3: Apply the power rule of Logarithm

According to the power rule of logarithms, logarithm base B of A to the power of N is equal to N times logarithm base B of A. The given logarithm can be rewritten as \(\log _{7^2} 7 = 2 \cdot \log_{7} 7\)

Step 4: Evaluate the result

Logarithm of a number to the same base equals 1 according to the identity rule of logarithms. So, \(\log_{7} 7 = 1\). Substitute this into our expression from the last step, the result will be \(2 \cdot 1\).