Simplify/Condense

Problem

1/2*(log_6)(x^4)+2+8*(log_6)(7−x)−2*(log_6)(x)

Solution

1. Apply the Power Property of logarithms, which states n*(log_b)(a)=(log_b)(a^n) to the first, third, and fourth terms.

1/2*(log_6)(x^4)=(log_6)((x^4)^(1/2))=(log_6)(x^2)

8*(log_6)(7−x)=(log_6)((7−x)^8)

2*(log_6)(x)=(log_6)(x^2)

2. Rewrite the constant 2 as a logarithm with base 6 using the identity n=(log_b)(b^n)

2=(log_6)(6^2)=(log_6)(36)

3. Substitute these expressions back into the original equation.

(log_6)(x^2)+(log_6)(36)+(log_6)((7−x)^8)−(log_6)(x^2)

4. Combine the logarithms using the Product Property (log_b)(M)+(log_b)(N)=(log_b)(M*N) and the Quotient Property (log_b)(M)−(log_b)(N)=(log_b)(M/N)

(log_6)(x^2⋅36⋅(7−x)^8)−(log_6)(x^2)

(log_6)((36*x^2*(7−x)^8)/(x^2))

5. Simplify the expression by canceling the common factor x^2 in the numerator and denominator.

(log_6)(36*(7−x)^8)

Final Answer

1/2*(log_6)(x^4)+2+8*(log_6)(7−x)−2*(log_6)(x)=(log_6)(36*(7−x)^8)

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