Question

solve the following elementary exponential equation.
$9^x - 3 = 0$
answer/how to enter your answer (opens in new window)
4 points
$x = $

Explanation:

Step1: Isolate the exponential term

We start with the equation \(9^{x}-3 = 0\). Add 3 to both sides of the equation to isolate the exponential term \(9^{x}\).

$$9^{x}=3$$

Step2: Express with the same base

We know that \(9 = 3^{2}\), so we can rewrite the left - hand side of the equation. Substitute \(9\) with \(3^{2}\) in the equation \(9^{x}=3\), we get \((3^{2})^{x}=3\).
Using the power - of - a - power rule \((a^{m})^{n}=a^{mn}\), the left - hand side becomes \(3^{2x}\). So our equation is now \(3^{2x}=3^{1}\).

Step3: Solve for x

Since the bases are the same (\(a^{m}=a^{n}\) implies \(m = n\) when \(a>0,a
eq1\)), and here \(a = 3\) which satisfies \(a>0,a
eq1\), we can set the exponents equal to each other.
So \(2x = 1\). Divide both sides of the equation by 2: \(x=\frac{1}{2}\)

Answer:

\(\frac{1}{2}\)