Let's solve the equation \( 9^x = \frac{1}{3} \times \sqrt{3} \) for \( x \) step-by-step.
### Step 1: Simplify the Right-Hand Side
First, let's simplify the expression on the right-hand side:
[tex]\[ \frac{1}{3} \times \sqrt{3} \][/tex]
The square root of 3 is approximately \( \sqrt{3} \approx 1.7320508075688772 \).
Therefore:
[tex]\[ \frac{1}{3} \times \sqrt{3} \approx \frac{1}{3} \times 1.7320508075688772 \approx 0.5773502691896257 \][/tex]
### Step 2: Rewrite the Equation
Rewrite the equation with the simplified right-hand side:
[tex]\[ 9^x = 0.5773502691896257 \][/tex]
### Step 3: Apply Logarithms
To solve for \( x \), we apply the logarithm on both sides. A common choice is the natural logarithm (ln), but any logarithm base can be used. We use the logarithm properties:
[tex]\[ x = \log_9(0.5773502691896257) \][/tex]
### Step 4: Change of Base Formula
We need to express \( \log_9 (0.5773502691896257) \) in terms of more commonly used logarithms. The change of base formula for logarithms is:
[tex]\[ \log_b(a) = \frac{\log_c(a)}{\log_c(b)} \][/tex]
where \( c \) can be any positive number, commonly we use base 10 (common logarithm) or base \( e \) (natural logarithm).
Using base \( e \) (natural logarithm), we get:
[tex]\[ x = \frac{\ln(0.5773502691896257)}{\ln(9)} \][/tex]
We know:
[tex]\[ \ln(0.5773502691896257) \approx -0.551 \][/tex]
[tex]\[ \ln(9) \approx 2.197 \][/tex]
Thus:
[tex]\[ x = \frac{-0.551}{2.197} \approx -0.25 \][/tex]
### Conclusion
The value of \( x \) that satisfies the equation \( 9^x = \frac{1}{3} \times \sqrt{3} \) is:
[tex]\[ x = -0.25 \][/tex]
So:
[tex]\[ x \approx -0.25 \][/tex]
