To solve the equation [tex]\( 1.13^x = 2.97 \)[/tex] for [tex]\( x \)[/tex], we can use the properties of logarithms. Specifically, we will use the fact that if [tex]\( a^x = b \)[/tex], then [tex]\( x = \frac{\log(b)}{\log(a)} \)[/tex].
Step-by-Step Solution:
1. Start with the given equation:
[tex]\[
1.13^x = 2.97
\][/tex]
2. Take the logarithm of both sides. It's most common to use the natural logarithm (ln), but common logarithms (log base 10) or any logarithm base will work.
[tex]\[
\log(1.13^x) = \log(2.97)
\][/tex]
3. Use the power rule of logarithms, which states that [tex]\(\log(a^b) = b \log(a)\)[/tex]:
[tex]\[
x \cdot \log(1.13) = \log(2.97)
\][/tex]
4. Solve for [tex]\( x \)[/tex] by isolating it:
[tex]\[
x = \frac{\log(2.97)}{\log(1.13)}
\][/tex]
So, the expression that Jackson can enter into his calculator to find [tex]\( x \)[/tex] is:
[tex]\[
x = \frac{\log(2.97)}{\log(1.13)}
\][/tex]
Using this logarithmic quotient on a calculator, Jackson will find that:
[tex]\[
x \approx 8.90675042995353
\][/tex]
Thus, the solution to the equation [tex]\( 1.13^x = 2.97 \)[/tex] is approximately [tex]\( x \approx 8.90675042995353 \)[/tex].
