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\\)
answer: how to enter your answer (opens in new window)
4 points
\\(x =\\)

Explanation:

Step1: Rewrite 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: Use the property of exponents

If \(a^m = a^n\), then \(m = n\) (for \(a>0,a
eq1\)). Here, \(a = 7\), so we have \(\log_{4}(x)=2\).

Step3: Convert logarithmic equation to exponential form

The logarithmic equation \(\log_{b}(y)=k\) is equivalent to \(b^k = y\). For \(\log_{4}(x)=2\), we have \(4^2=x\).

Step4: Calculate \(4^2\)

\(4^2 = 16\), so \(x = 16\).

Answer:

\(x=\frac{16}{1}\) (or simply \(16\))