Question

1. find the exact value of the logarithm without using a calculator.

\log_{\sqrt{7}} 49
\log_{\sqrt{7}} 49 = \square

Explanation:

Step1: Recall the logarithm definition

Let \( \log_{a}b = x \), which means \( a^{x}=b \). For \( \log_{\sqrt{7}} 49 = x \), we have \( (\sqrt{7})^{x}=49 \).

Step2: Express numbers as powers of 7

We know that \( \sqrt{7}=7^{\frac{1}{2}} \) and \( 49 = 7^{2} \). Substitute these into the equation: \( (7^{\frac{1}{2}})^{x}=7^{2} \).

Step3: Use exponent rules

Using the power - of - a - power rule \( (a^{m})^{n}=a^{mn} \), the left - hand side becomes \( 7^{\frac{x}{2}} \). So we have the equation \( 7^{\frac{x}{2}}=7^{2} \).
Since the bases are the same and the exponential function \( y = a^{x}(a>0,a
eq1) \) is one - to - one, we can set the exponents equal to each other: \( \frac{x}{2}=2 \).

Step4: Solve for x

Multiply both sides of the equation \( \frac{x}{2}=2 \) by 2. We get \( x = 4 \).

Answer:

\( 4 \)