Question

solve. write your answer as an integer or a fraction in simplest form.
81^x = 3
x =

Explanation:

Step1: Express 81 as a power of 3

We know that \( 81 = 3^4 \), so we can rewrite the left - hand side of the equation \( 81^x=3 \) as \( (3^4)^x \).
According to the power - of - a - power rule \((a^m)^n=a^{m\times n}\), we have \((3^4)^x = 3^{4x}\). So the equation becomes \(3^{4x}=3^1\).

Step2: Solve for x

Since the bases are the same (\(a^m=a^n\) implies \(m = n\) when \(a>0,a
eq1\)), and here \(a = 3\), we can set the exponents equal to each other: \(4x=1\).
To solve for \(x\), we divide both sides of the equation by 4: \(x=\frac{1}{4}\).

Answer:

\(\frac{1}{4}\)