To solve for [tex]\( z \)[/tex] in the equation [tex]\( 1.13^z = 2.97 \)[/tex], we can follow these steps:
1. Take the natural logarithm of both sides of the equation to make the exponent more manageable:
[tex]\[
\ln(1.13^z) = \ln(2.97)
\][/tex]
2. Apply the power rule of logarithms which states that [tex]\(\ln(a^b) = b \cdot \ln(a)\)[/tex]. This allows us to move the exponent [tex]\( z \)[/tex] in front of the logarithm:
[tex]\[
z \cdot \ln(1.13) = \ln(2.97)
\][/tex]
3. Isolate [tex]\(z\)[/tex] by dividing both sides of the equation by [tex]\(\ln(1.13)\)[/tex]:
[tex]\[
z = \frac{\ln(2.97)}{\ln(1.13)}
\][/tex]
Jackson can now enter this expression into his calculator to find the value of [tex]\( z \)[/tex]. The correct logarithmic quotient is:
[tex]\[
\boxed{\frac{\ln(2.97)}{\ln(1.13)}}
\][/tex]
