Let's solve the equation step-by-step.
The given equation is:
[tex]\[
\ln 20 + \ln 5 = 2 \ln x
\][/tex]
### Step 1: Combine the logarithmic terms on the left-hand side
We can use the property of logarithms that [tex]\(\ln a + \ln b = \ln(ab)\)[/tex] to combine the logarithms on the left side:
[tex]\[
\ln(20 \cdot 5) = 2 \ln x
\][/tex]
### Step 2: Simplify the combined logarithmic term
Next, multiply the numbers inside the logarithm:
[tex]\[
\ln(100) = 2 \ln x
\][/tex]
### Step 3: Use the property of logarithms to simplify further
We can use the property that [tex]\(k \ln a = \ln(a^k)\)[/tex]. Here, [tex]\(k = 2\)[/tex]:
[tex]\[
\ln(100) = \ln(x^2)
\][/tex]
### Step 4: Equate the arguments of the logarithms
If [tex]\(\ln(a) = \ln(b)\)[/tex], then [tex]\(a = b\)[/tex]. Therefore, we equate the arguments of the logarithms:
[tex]\[
100 = x^2
\][/tex]
### Step 5: Solve for [tex]\(x\)[/tex]
To find [tex]\(x\)[/tex], we take the square root of both sides:
[tex]\[
x = \sqrt{100}
\][/tex]
[tex]\[
x = 10
\][/tex]
So, the solution to the equation [tex]\(\ln 20 + \ln 5 = 2 \ln x\)[/tex] is:
[tex]\[
x = 10
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{x = 10}
\][/tex]
