To solve the exponential equation [tex]\(11e^x = 4\)[/tex] for [tex]\(x\)[/tex], follow these steps:
1. Isolate the exponential expression: Divide both sides of the equation by 11 to isolate the term involving the exponent.
[tex]\[
e^x = \frac{4}{11}
\][/tex]
2. Apply the natural logarithm to both sides: The natural logarithm, denoted as [tex]\(\ln\)[/tex], is the inverse of the exponential function. Taking the natural logarithm of both sides will help to solve for [tex]\(x\)[/tex].
[tex]\[
\ln(e^x) = \ln\left(\frac{4}{11}\right)
\][/tex]
3. Simplify using the property of logarithms: The natural logarithm of [tex]\(e^x\)[/tex] is simply [tex]\(x\)[/tex], because [tex]\(\ln(e^x) = x\ln(e) = x\)[/tex].
[tex]\[
x = \ln\left(\frac{4}{11}\right)
\][/tex]
4. Calculate the value: Using a calculator to find the natural logarithm of [tex]\(\frac{4}{11}\)[/tex], we get:
[tex]\[
x \approx -1.0116
\][/tex]
Thus, the solution to the equation [tex]\(11e^x = 4\)[/tex] is:
[tex]\[
x \approx -1.0116
\][/tex]
