To solve for [tex]\(x\)[/tex] in the equation [tex]\(22^x = 17\)[/tex], we can follow these steps:
1. Take the natural logarithm (ln) on both sides of the equation:
[tex]\[
\ln(22^x) = \ln(17)
\][/tex]
2. Use the logarithm power rule which states that [tex]\(\ln(a^b) = b \cdot \ln(a)\)[/tex]. Applying this rule, we get:
[tex]\[
x \cdot \ln(22) = \ln(17)
\][/tex]
3. Solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{\ln(17)}{\ln(22)}
\][/tex]
So, the value of [tex]\(x\)[/tex] is found by dividing the natural logarithm of 17 by the natural logarithm of 22.
Using this approach, the calculated value of [tex]\(x\)[/tex] is:
[tex]\[
x \approx 0.9165882988692126
\][/tex]
Therefore, the value of [tex]\(x\)[/tex] that satisfies the equation [tex]\(22^x = 17\)[/tex] is approximately 0.9165882988692126.
