Let's solve the equation step by step:

$2 \log 50 = 3 \log 25 + \log (x - 2)$

Step 1: Use the power rule of logarithms

The power rule states that $a \log b = \log(b^a)$. We will apply this to simplify the terms on both sides.

• On the left side: $2 \log 50 = \log (50^2) = \log 2500$

• On the right side: $3 \log 25 = \log (25^3) = \log 15625$

So the equation becomes: $\log 2500 = \log 15625 + \log (x - 2)$

Step 2: Apply the logarithmic property for addition

The property $\log a + \log b = \log(ab)$ allows us to combine the terms on the right-hand side:

$\log 2500 = \log (15625(x - 2))$

Step 3: Eliminate the logarithms

Since the logarithms are equal on both sides, we can drop the $\log$ and set the arguments equal:

$2500 = 15625(x - 2)$

Step 4: Solve for $x$

Now, solve the equation for $x$.

1. Divide both sides by 15625: $\frac{2500}{15625} = x - 2$

2. Simplify the fraction: $\frac{2500}{15625} = \frac{2}{12.5} = 0.16$ Thus, we have: $0.16 = x - 2$

3. Add 2 to both sides: $x = 2 + 0.16 = 2.16$

Final Answer:

$x = 2.16$

Let me know if you'd like any further details or have any questions!

Related Questions:

1. How do logarithmic properties apply when solving for variables inside a logarithmic equation?
2. How do you handle logarithmic equations with different bases (e.g., base 10 vs. base $e$)?
3. What are the steps to solve an exponential equation like $2^x = 16$?
4. How can you verify your solution to a logarithmic equation?
5. What is the inverse relationship between logarithms and exponents?

Tip: Always remember that logarithmic properties (like the power rule and the product rule) simplify complex equations, making them easier to solve!