To solve the expression $\log_5(125x^3)$, we'll break it down step by step using logarithmic properties.

Step-by-Step Solution

1. Express 125 as a Power of 5: The number 125 can be expressed as $5^3$. So: $125x^3 = (5^3) \times x^3$

2. Rewrite the Logarithm: Substitute back into the logarithmic expression: $\log_5(125x^3) = \log_5((5^3) \times x^3)$

3. Apply the Product Rule for Logarithms: Recall that $\log_a(bc) = \log_a(b) + \log_a(c)$: $\log_5((5^3) \times x^3) = \log_5(5^3) + \log_5(x^3)$

4. Simplify Each Logarithm:

   • For $\log_5(5^3)$, apply the power rule $\log_a(a^b) = b$: $\log_5(5^3) = 3$
   • For $\log_5(x^3)$, also apply the power rule $\log_a(b^c) = c \cdot \log_a(b)$: $\log_5(x^3) = 3 \log_5(x)$

5. Combine the Results: $\log_5(125x^3) = 3 + 3\log_5(x)$

Final Answer:

$\log_5(125x^3) = 3 + 3\log_5(x)$

Would you like more details on any part of this solution, or do you have any questions?

Related Questions to Expand Understanding:

1. How would you solve $\log_5(25x^2)$?
2. What is the result of $\log_5(625)$?
3. How do you apply the power rule to simplify $\log_5(x^4)$?
4. Can you explain the properties of logarithms, such as the product rule and power rule, in more depth?
5. How would you solve $\log_5(5x^5) + \log_5(x^3)$?

Tip:

Always try to express numbers in the logarithm as powers of the base if possible—this helps simplify the calculation significantly.