Expand the Logarithmic Expression log base 6 of (ab^4)/( cube root of c^2)

Problem

(log_6)((a*b4)/√(3,c2))

Solution

1. Apply the quotient rule for logarithms, which states that (log_b)(x/y)=(log_b)(x)−(log_b)(y)

(log_6)(a*b4)−(log_6)(√(3,c2))

2. Apply the product rule to the first term, which states that (log_b)(x*y)=(log_b)(x)+(log_b)(y)

(log_6)(a)+(log_6)(b4)−(log_6)(√(3,c2))

3. Rewrite the radical using a rational exponent, where √(n,xm)=x(m/n)

(log_6)(a)+(log_6)(b4)−(log_6)(c(2/3))

4. Apply the power rule for logarithms, which states that (log_b)(xp)=p*(log_b)(x) to the second and third terms.

(log_6)(a)+4*(log_6)(b)−2/3*(log_6)(c)

Final Answer

(log_6)((a*b4)/√(3,c2))=(log_6)(a)+4*(log_6)(b)−2/3*(log_6)(c)

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