Question

x = log₉(3)

Explanation:

Step1: Recall logarithm definition

If \( y = \log_{b}(a) \), then \( b^{y}=a \). Here, \( x = \log_{9}(3) \), so \( 9^{x}=3 \).

Step2: Express 9 as power of 3

Since \( 9 = 3^{2} \), substitute into equation: \( (3^{2})^{x}=3 \).

Step3: Simplify left - hand side

Using exponent rule \( (a^{m})^{n}=a^{mn} \), we get \( 3^{2x}=3^{1} \).

Step4: Set exponents equal

If \( a^{m}=a^{n} \) and \( a>0,a
eq1 \), then \( m = n \). So \( 2x = 1 \).

Step5: Solve for x

Divide both sides by 2: \( x=\frac{1}{2} \).

Answer:

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