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The equation to solve is:

\[
10^{5x + 2} = 20
\]

Let's solve it step by step:

### Step 1: Isolate the exponential term
First, divide both sides of the equation by \(10^2\):

\[
10^{5x+2} = 20
\]
\[
\frac{10^{5x+2}}{10^2} = \frac{20}{10^2}
\]
\[
10^{5x} = \frac{20}{100} = 0.2
\]

### Step 2: Take the logarithm of both sides
Now, take the logarithm (base 10) of both sides:

\[
\log(10^{5x}) = \log(0.2)
\]
Since \(\log(10^{5x}) = 5x\), this simplifies to:

\[
5x = \log(0.2)
\]

### Step 3: Solve for \(x\)
Now solve for \(x\):

\[
x = \frac{\log(0.2)}{5}
\]

### Step 4: Simplify \(\log(0.2)\)
We know that \(\log(0.2) = \log\left(\frac{1}{5}\right) = -\log(5)\), so:

\[
x = \frac{-\log(5)}{5}
\]

This matches the second option in the list:

\[
x = \frac{\log(20) - 2}{5}
\]

So, the correct answer is the second option: 
\[
x = \frac{\log(20) - 2}{5}
\]

Would you like further details on this, or have any additional questions?

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Here are 5 related questions:
1. What are the properties of logarithms that help solve exponential equations?
2. How would the solution change if the base of the exponent were not 10?
3. Can you solve the equation using natural logarithms (\(\ln\)) instead of common logarithms (\(\log\))?
4. What is the graphical interpretation of solving this equation?
5. How do you apply logarithms to solve similar equations with different bases?

**Tip:** When solving exponential equations, taking the logarithm of both sides is an effective method to simplify and solve for the exponent.

Learn how to solve the exponential equation 10^(5x + 2) = 20 using logarithms. This problem covers key concepts like taking logarithms of both sides to simplify exponential terms. Suitable for students in Grades 9-12.

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