To solve the equation [tex]\(3^{2z} + 12 = 39\)[/tex], we follow these steps:
1. Isolate the exponential term:
[tex]\[
3^{2z} + 12 = 39
\][/tex]
Subtract 12 from both sides:
[tex]\[
3^{2z} = 27
\][/tex]
2. Rewrite the equation using logarithms to solve for [tex]\(2z\)[/tex]:
To bring the exponent [tex]\(2z\)[/tex] down, we can take the natural logarithm ([tex]\(\ln\)[/tex]) of both sides:
[tex]\[
\ln(3^{2z}) = \ln(27)
\][/tex]
Using the power rule of logarithms, [tex]\(\ln(a^b) = b \cdot \ln(a)\)[/tex]:
[tex]\[
2z \cdot \ln(3) = \ln(27)
\][/tex]
3. Solve for [tex]\(z\)[/tex]:
Divide both sides by [tex]\(2 \cdot \ln(3)\)[/tex]:
[tex]\[
z = \frac{\ln(27)}{2 \cdot \ln(3)}
\][/tex]
4. Evaluate the expression:
We know that [tex]\( \ln(27) = \ln(3^3) = 3 \cdot \ln(3) \)[/tex]. Therefore:
[tex]\[
z = \frac{3 \cdot \ln(3)}{2 \cdot \ln(3)} = \frac{3}{2}
\][/tex]
Therefore, the correct answer is [tex]\( z = \frac{3}{2} \)[/tex], which corresponds to option B.
So, the correct answer is:
B. [tex]\( z = \frac{3}{2} \)[/tex]
