To solve the exponential equation [tex]\(37^x = 12\)[/tex], follow these detailed steps:
1. Start with the given equation:
[tex]\[
37^x = 12
\][/tex]
2. Take the natural logarithm (or log base 10) of both sides of the equation:
[tex]\[
\log(37^x) = \log(12)
\][/tex]
3. Apply the power rule of logarithms which states that [tex]\(\log(a^b) = b \cdot \log(a)\)[/tex]:
[tex]\[
x \cdot \log(37) = \log(12)
\][/tex]
4. Isolate the variable [tex]\(x\)[/tex] by dividing both sides of the equation by [tex]\(\log(37)\)[/tex]:
[tex]\[
x = \frac{\log(12)}{\log(37)}
\][/tex]
5. Evaluate the logarithms to find their numerical values. Using a calculator or logarithm tables:
[tex]\[
\log(37) \approx 3.6109179126442243
\][/tex]
[tex]\[
\log(12) \approx 2.4849066497880004
\][/tex]
6. Substitute these values back into the equation to solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{\log(12)}{\log(37)} \approx \frac{2.4849066497880004}{3.6109179126442243}
\][/tex]
7. Perform the division:
[tex]\[
x \approx 0.6881648129099501
\][/tex]
Thus, the solution to the equation [tex]\(37^x = 12\)[/tex] is:
[tex]\[
x \approx 0.688
\][/tex]
In conclusion, to solve the exponential equation [tex]\(37^x = 12\)[/tex], you need to take the natural logarithm of both sides, use the power rule of logarithms, isolate the variable [tex]\(x\)[/tex], and then evaluate the logarithms to find the numerical solution [tex]\(x \approx 0.688\)[/tex].
