Answer :
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]
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]
