Simplify/Condense log base 6 of (5)^4+ log base 6 of 25

Problem

(log_6)(5)+(log_6)(25)

Solution

1. Identify the logarithmic expression and recognize that both terms have the same base, which is 6

2. Rewrite the second term by expressing 25 as a power of 5 to make the terms more consistent.

(log_6)(25)=(log_6)(5)

3. Apply the product rule for logarithms, which states that (log_b)(M)+(log_b)(N)=(log_b)(M⋅N)

(log_6)(5)+(log_6)(5)=(log_6)(5⋅5)

4. Simplify the argument inside the logarithm by using the product rule for exponents, am⋅an=a(m+n)

5⋅5=5(4+2)=5

5. Substitute the simplified argument back into the logarithmic expression.

(log_6)(5)

6. Apply the power rule for logarithms, (log_b)(Mp)=p*(log_b)(M) to move the exponent in front of the log.

(log_6)(5)=6*(log_6)(5)

Final Answer

(log_6)(5)+(log_6)(25)=6*(log_6)(5)

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