Expand the Logarithmic Expression log base 5 of 125x

Problem

(log_5)(125*x)

Solution

1. Identify the logarithmic property for multiplication, which states that the log of a product is the sum of the logs: (log_b)(M*N)=(log_b)(M)+(log_b)(N)

2. Apply the property to the expression by splitting the product 125*x into two separate logarithmic terms.

(log_5)(125*x)=(log_5)(125)+(log_5)(x)

3. Evaluate the numerical term (log_5)(125) by determining what power 5 must be raised to in order to equal 125

5=125

(log_5)(125)=3

4. Substitute the evaluated constant back into the expanded expression.

3+(log_5)(x)

Final Answer

(log_5)(125*x)=3+(log_5)(x)

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