To solve the equation [tex]\(-3 \cdot e^{5w} = -88\)[/tex] for [tex]\(w\)[/tex], follow these steps:
1. Isolate the exponential term:
[tex]\[
-3 \cdot e^{5w} = -88
\][/tex]
Divide both sides by [tex]\(-3\)[/tex]:
[tex]\[
e^{5w} = \frac{88}{3}
\][/tex]
2. Take the natural logarithm of both sides:
[tex]\[
\ln(e^{5w}) = \ln\left(\frac{88}{3}\right)
\][/tex]
Using the property of logarithms [tex]\(\ln(e^x) = x\)[/tex], this simplifies to:
[tex]\[
5w = \ln\left(\frac{88}{3}\right)
\][/tex]
3. Solve for [tex]\(w\)[/tex]:
Divide both sides by 5:
[tex]\[
w = \frac{\ln\left(\frac{88}{3}\right)}{5}
\][/tex]
Hence, the exact solution as a logarithm in base-e is:
[tex]\[
w = \frac{\ln\left(\frac{88}{3}\right)}{5}
\][/tex]
To approximate the value of [tex]\(w\)[/tex], evaluate the logarithm and then divide by 5. The numerical result is approximately:
[tex]\[
w \approx 0.676
\][/tex]
Thus, the approximate value of [tex]\(w\)[/tex] rounded to the nearest thousandth is:
[tex]\[
w \approx 0.676
\][/tex]
