Question

use the elementary properties of logarithms to solve the following equation. write your answer as a fraction reduced to lowest terms.
$7^{\log_{4}(x)} = 49$
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Explanation:

Step1: Express 49 as a power of 7

We know that \(49 = 7^2\). So the equation \(7^{\log_{4}(x)} = 49\) can be rewritten as \(7^{\log_{4}(x)} = 7^2\).

Step2: Set the exponents equal

Since the bases are the same (both are 7) and the exponential function is one - to - one, we can set the exponents equal to each other. So we have \(\log_{4}(x)=2\).

Step3: Convert the logarithmic equation to exponential form

Recall that the logarithmic equation \(\log_{a}(b)=c\) is equivalent to the exponential equation \(a^{c}=b\). For \(\log_{4}(x) = 2\), where \(a = 4\), \(c=2\) and \(b=x\), we get \(x = 4^{2}\).

Step4: Calculate \(4^{2}\)

\(4^{2}=16\).

Answer:

\(16\)