To solve the equation [tex]\(\ln 20 + \ln 5 = 2 \ln x\)[/tex], let's follow these steps:
1. Combine the logarithmic terms on the left side using the product rule of logarithms. The product rule states that [tex]\(\ln a + \ln b = \ln (ab)\)[/tex]. Therefore, we have:
[tex]\[
\ln 20 + \ln 5 = \ln (20 \cdot 5)
\][/tex]
2. Multiply the numbers inside the logarithm:
[tex]\[
\ln (20 \cdot 5) = \ln 100
\][/tex]
3. Rewrite the equation in terms of a single logarithm on both sides:
[tex]\[
\ln 100 = 2 \ln x
\][/tex]
4. Use the power rule of logarithms to simplify the right side. The power rule states that [tex]\(k \ln a = \ln (a^k)\)[/tex]. Therefore, we get:
[tex]\[
\ln 100 = \ln (x^2)
\][/tex]
5. If the logarithms are equal, then their arguments must also be equal. So we equate the arguments:
[tex]\[
100 = x^2
\][/tex]
6. Solve for [tex]\(x\)[/tex] by taking the square root of both sides. The square root of [tex]\(100\)[/tex] is [tex]\(10\)[/tex]:
[tex]\[
x = \sqrt{100} = 10
\][/tex]
So the solution is [tex]\(x = 10\)[/tex].
Thus, the correct answer is [tex]\(x = 10\)[/tex].
