To solve the equation [tex]\(24 \cdot \log(3x) = 60\)[/tex], follow these steps:
1. Isolate [tex]\(\log(3x)\)[/tex]:
[tex]\[
24 \cdot \log(3x) = 60
\][/tex]
Divide both sides by 24:
[tex]\[
\log(3x) = \frac{60}{24}
\][/tex]
Simplify the fraction:
[tex]\[
\log(3x) = 2.5
\][/tex]
2. Convert the logarithmic equation to an exponential form:
The definition of logarithms tells us that if [tex]\(\log_b(a) = c\)[/tex], then [tex]\(a = b^c\)[/tex]. Here, we're dealing with a common logarithm (base 10):
[tex]\[
3x = 10^{2.5}
\][/tex]
3. Calculate [tex]\(10^{2.5}\)[/tex]:
Using the property of exponents:
[tex]\[
10^{2.5} = 10^{2 + 0.5} = 10^2 \cdot 10^{0.5}
\][/tex]
[tex]\[
10^{2.5} = 100 \cdot \sqrt{10} \approx 100 \cdot 3.1623 = 316.23
\][/tex]
Therefore,
[tex]\[
3x = 316.23
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
Divide both sides by 3:
[tex]\[
x = \frac{316.23}{3} \approx 105.41
\][/tex]
So, the solution to the equation [tex]\(24 \cdot \log(3x) = 60\)[/tex] is:
[tex]\[
\boxed{105.41}
\][/tex]
Thus, the correct answer is:
D. [tex]\(x = 105.41\)[/tex]
