To solve the equation [tex]\( 3^x = 43 \)[/tex], we need to find the value of [tex]\( x \)[/tex] that satisfies this exponential equation.
Here are the steps to solve this problem:
1. Understanding the Exponential Equation:
[tex]\[
3^x = 43
\][/tex]
2. Taking the Natural Logarithm on Both Sides:
Taking the natural logarithm (ln) on both sides helps us to bring the exponent down and solve for [tex]\( x \)[/tex]:
[tex]\[
\ln(3^x) = \ln(43)
\][/tex]
3. Using Logarithm Properties:
Use the property of logarithms that states [tex]\( \ln(a^b) = b \ln(a) \)[/tex]:
[tex]\[
x \ln(3) = \ln(43)
\][/tex]
4. Isolate [tex]\( x \)[/tex]:
Solve for [tex]\( x \)[/tex] by dividing both sides by [tex]\( \ln(3) \)[/tex]:
[tex]\[
x = \frac{\ln(43)}{\ln(3)}
\][/tex]
5. Calculation:
Now we calculate the values:
[tex]\[
\ln(43) \approx 3.7612 \quad \text{and} \quad \ln(3) \approx 1.0986
\][/tex]
Therefore,
[tex]\[
x = \frac{3.7612}{1.0986} \approx 3.423
\][/tex]
After these steps, we find that the solution to the equation [tex]\( 3^x = 43 \)[/tex] is approximately [tex]\( x \approx 3.423 \)[/tex].
