To solve the exponential equation [tex]\( 3^x = 6 \)[/tex], we will use logarithms to find the value of [tex]\( x \)[/tex]. Here's a step-by-step solution:
1. Identify the equation: We start with the equation given:
[tex]\[
3^x = 6
\][/tex]
2. Take the logarithm of both sides: To solve for [tex]\( x \)[/tex], we will take the natural logarithm (ln) or logarithm (log) of both sides of the equation. This helps us use the properties of logarithms to bring the exponent down. Here, we will use the natural logarithm:
[tex]\[
\ln(3^x) = \ln(6)
\][/tex]
3. Apply the power rule of logarithms: The power rule of logarithms states that [tex]\(\ln(a^b) = b \cdot \ln(a)\)[/tex]. Using this rule, we can bring the exponent [tex]\( x \)[/tex] in front:
[tex]\[
x \cdot \ln(3) = \ln(6)
\][/tex]
4. Isolate [tex]\( x \)[/tex]: To isolate [tex]\( x \)[/tex], we divide both sides of the equation by [tex]\(\ln(3)\)[/tex]:
[tex]\[
x = \frac{\ln(6)}{\ln(3)}
\][/tex]
5. Calculate the value: Using a calculator to compute the natural logarithms and their division, we find:
[tex]\[
x \approx \frac{\ln(6)}{\ln(3)} \approx 1.631
\][/tex]
Therefore, the solution to the equation [tex]\( 3^x = 6 \)[/tex] is:
[tex]\[
x = 1.631
\][/tex]
So, the correct choice is:
A. [tex]\( x = 1.631 \)[/tex]
