Full solution

9^{x}=36 \quad x=?

Identify Base and Target: Identify the base and the target value in the equation $9^{x} = 36$ . $\newline$ Base: $9$ $\newline$ Target value: $36$
Recognize Perfect Squares: Recognize that both $9$ and $36$ are perfect squares. $9$ is $3^2$ and $36$ is $6^2$ .
Rewrite Using Square Roots: Rewrite the equation using the square roots: $(3^2)^x = (6^2)$ .
Apply Power of Power Rule: Apply the power of a power rule: $(a^m)^n = a^{(m*n)}$ . So, $(3^2)^x = 3^{(2x)}$ .
Rewrite with New Bases: Rewrite the equation with the new bases: $3^{2x} = 6^2$ .
Equation with Same Base: Since $6^2$ can be written as $(3^2)^1$ , we have $3^{2x} = (3^2)^1$ .
Apply Exponents Equality: Apply the power of a power rule again: $(3^2)^1 = 3^{2*1} = 3^2$ .
Solve for x: Now we have an equation with the same base: $3^{2x} = 3^2$ .
Calculate $x$ : Since the bases are the same, we can set the exponents equal to each other: $2x = 2$ .
Calculate $x$ : Since the bases are the same, we can set the exponents equal to each other: $2x = 2$ .Divide both sides of the equation by $2$ to solve for $x$ : $x = \frac{2}{2}$ .
Calculate $x$ : Since the bases are the same, we can set the exponents equal to each other: $2x = 2$ .Divide both sides of the equation by $2$ to solve for $x$ : $x = \frac{2}{2}$ .Calculate the value of $x$ : $x = 1$ .