Simplify/Condense log base 5 of 150

Problem

(log_5)(150)

Solution

1. Identify the factors of the argument 150 that are powers of the base 5

2. Factor the number 150 into a product of its prime factors or multiples of 5

150=25⋅6

150=5⋅6

3. Apply the product rule for logarithms, which states (log_b)(x*y)=(log_b)(x)+(log_b)(y)

(log_5)(5⋅6)=(log_5)(5)+(log_5)(6)

4. Simplify the term (log_5)(5) using the property (log_b)(bx)=x

(log_5)(5)=2

5. Combine the results to write the condensed or simplified expression.

2+(log_5)(6)

Final Answer

(log_5)(150)=2+(log_5)(6)

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