To determine the value of [tex]\( x \)[/tex] in the equation [tex]\( 1.13^x = 2.97 \)[/tex], you can follow these steps:
1. Rewrite the equation using logarithms:
Start with the given equation:
[tex]\[
1.13^x = 2.97
\][/tex]
To solve for [tex]\( x \)[/tex], take the logarithm of both sides of the equation. You can use any logarithm, but for simplicity, we'll use the natural logarithm (ln):
[tex]\[
\ln(1.13^x) = \ln(2.97)
\][/tex]
2. Apply the power rule of logarithms:
The power rule of logarithms states that [tex]\( \ln(a^b) = b \cdot \ln(a) \)[/tex]. Applying this rule to the left side of the equation gives:
[tex]\[
x \cdot \ln(1.13) = \ln(2.97)
\][/tex]
3. Isolate [tex]\( x \)[/tex]:
To solve for [tex]\( x \)[/tex], divide both sides of the equation by [tex]\( \ln(1.13) \)[/tex]:
[tex]\[
x = \frac{\ln(2.97)}{\ln(1.13)}
\][/tex]
By entering this logarithmic quotient into a calculator, Jackson can find the value of [tex]\( x \)[/tex] with high precision.
The numerical result is:
[tex]\[
x \approx 8.90675042995353
\][/tex]
So, the step-by-step expression Jackson needs to enter into his calculator is:
[tex]\[
x = \frac{\ln(2.97)}{\ln(1.13)}
\][/tex]
