To solve the given exponential equation [tex]\(37^x = 12\)[/tex], we need to isolate the variable [tex]\(x\)[/tex]. Here is a step-by-step guide to solving the problem:
1. Understand the problem: We are given the equation in the form of [tex]\(a^x = b\)[/tex], where [tex]\(a = 37\)[/tex] and [tex]\(b = 12\)[/tex]. We need to solve for [tex]\(x\)[/tex].
2. Take the logarithm of both sides: To isolate [tex]\(x\)[/tex], we can take the logarithm of both sides of the equation. This step helps us to deal with the exponent. We can use any logarithm base (such as natural log, ln, or common log, log), but for this explanation, we will use the natural logarithm (ln):
[tex]\[
\ln(37^x) = \ln(12)
\][/tex]
3. Use the power rule of logarithms: The power rule says that [tex]\(\ln(a^x) = x \cdot \ln(a)\)[/tex]. Applying this rule allows us to bring the exponent [tex]\(x\)[/tex] in front of the logarithm:
[tex]\[
x \cdot \ln(37) = \ln(12)
\][/tex]
4. Isolate [tex]\(x\)[/tex]: Now, solve for [tex]\(x\)[/tex] by dividing both sides of the equation by [tex]\(\ln(37)\)[/tex]:
[tex]\[
x = \frac{\ln(12)}{\ln(37)}
\][/tex]
5. Calculate the result: Finally, you compute the value of the natural logarithms on both sides and divide them:
[tex]\[
x \approx \frac{\ln(12)}{\ln(37)} \approx 0.688
\][/tex]
So, the solution to the equation [tex]\(37^x = 12\)[/tex] is approximately [tex]\(x = 0.688\)[/tex]. Therefore, we find that [tex]\(x \approx 0.688\)[/tex]. This means that raising 37 to the power of approximately 0.688 results in the value 12.
