To solve the equation [tex]\( 3^x = 23 \)[/tex] for [tex]\( x \)[/tex]:
### Step-by-Step Solution:
1. Rewrite the equation in logarithmic form:
The equation [tex]\( 3^x = 23 \)[/tex] can be expressed using logarithms. To isolate [tex]\( x \)[/tex], take the natural logarithm (log base [tex]\( e \)[/tex], denoted as [tex]\(\ln\)[/tex]) on both sides of the equation:
[tex]\[
\ln(3^x) = \ln(23)
\][/tex]
2. Use the power rule of logarithms:
According to the power rule, [tex]\(\ln(a^b) = b \cdot \ln(a)\)[/tex]. Apply this rule to simplify the left-hand side:
[tex]\[
x \cdot \ln(3) = \ln(23)
\][/tex]
3. Isolate [tex]\( x \)[/tex]:
Solve for [tex]\( x \)[/tex] by dividing both sides of the equation by [tex]\(\ln(3)\)[/tex]:
[tex]\[
x = \frac{\ln(23)}{\ln(3)}
\][/tex]
### Exact Solution:
[tex]\[
x = \frac{\ln(23)}{\ln(3)}
\][/tex]
### Approximate Numerical Solution:
To find the approximate value of [tex]\( x \)[/tex] to 4 decimal places, we perform the division:
[tex]\[
x \approx 2.854049830200271
\][/tex]
Rounding to four decimal places:
[tex]\[
x \approx 2.854
\][/tex]
### Final Answers:
- The exact solution is:
[tex]\[
x = \frac{\ln(23)}{\ln(3)}
\][/tex]
- The solution to 4 decimal places is:
[tex]\[
x \approx 2.854
\][/tex]
