To solve $5 \log 625^3$, we can break it down step by step using logarithmic properties.

Step 1: Apply the power rule of logarithms

The power rule states that $\log_b (a^n) = n \log_b a$. So, we can take the exponent 3 in the logarithm outside as a multiplier: $5 \log 625^3 = 5 \times 3 \log 625$ This simplifies to: $5 \log 625^3 = 15 \log 625$

Step 2: Express 625 as a power of 5

We know that $625 = 5^4$. So we can rewrite the logarithmic expression: $\log 625 = \log (5^4)$

Step 3: Apply the power rule again

Using the power rule $\log_b (a^n) = n \log_b a$ again, we get: $\log (5^4) = 4 \log 5$

Step 4: Substituting back into the expression

Now substitute $\log 625 = 4 \log 5$ into the equation: $15 \log 625 = 15 \times 4 \log 5 = 60 \log 5$

Final Answer:

The expression simplifies to: $5 \log 625^3 = 60 \log 5$

Would you like to go further into this or have any more questions?

Here are 5 related questions you can explore:

1. What is the value of $\log 5$ in base 10?
2. How do logarithmic rules apply to logarithms of fractions?
3. What are the properties of logarithms that simplify expressions with products or quotients?
4. How can you simplify logarithmic expressions with different bases?
5. What are the common logarithms and natural logarithms used for?

Tip: Always apply logarithmic rules systematically: power rule, product rule, and change of base rule help simplify complex expressions.