To solve the equation [tex]\( 3 \cdot 5^{0.2w} = 720 \)[/tex], we can follow these steps:
1. Isolate the exponential term: We first want to isolate [tex]\( 5^{0.2w} \)[/tex] on one side of the equation. We do this by dividing both sides of the equation by 3:
[tex]\[
5^{0.2w} = \frac{720}{3}
\][/tex]
Simplifying the right side:
[tex]\[
5^{0.2w} = 240
\][/tex]
2. Take the natural logarithm of both sides: By taking the natural logarithm (ln) of both sides of the equation, we can employ the property of logarithms that allows us to bring the exponent down:
[tex]\[
\ln(5^{0.2w}) = \ln(240)
\][/tex]
Using the logarithm property [tex]\( \ln(a^b) = b \cdot \ln(a) \)[/tex]:
[tex]\[
0.2w \cdot \ln(5) = \ln(240)
\][/tex]
3. Solve for [tex]\( w \)[/tex]: To isolate [tex]\( w \)[/tex], we divide both sides of the equation by [tex]\( 0.2 \cdot \ln(5) \)[/tex]:
[tex]\[
w = \frac{\ln(240)}{0.2 \cdot \ln(5)}
\][/tex]
Substituting the values of [tex]\( \ln(240) \)[/tex] and [tex]\( \ln(5) \)[/tex]:
- [tex]\( \ln(240) \approx 5.480638923341991 \)[/tex]
- [tex]\( \ln(5) \approx 1.6094379124341003 \)[/tex]
So, we get:
[tex]\[
w = \frac{5.480638923341991}{0.2 \cdot 1.6094379124341003}
\][/tex]
Calculating the denominator:
[tex]\[
0.2 \cdot 1.6094379124341003 \approx 0.3218875824868201
\][/tex]
Now, dividing the numerator by the denominator:
[tex]\[
w \approx 17.026562133897787
\][/tex]
4. Round the answer: Finally, we round the result to the nearest thousandth:
[tex]\[
w \approx 17.027
\][/tex]
So, the solution to the equation [tex]\( 3 \cdot 5^{0.2w} = 720 \)[/tex] is:
[tex]\[
w \approx 17.027
\][/tex]
