Expand the Logarithmic Expression log base 8 of 4a((b-4)/(c^4))

Problem

(log_8)((4*a*(b−4))/(c4))

Solution

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

(log_8)(4*a)+(log_8)((b−4)/(c4))

2. Apply the product rule again to the first term to separate the constant and the variable.

(log_8)(4)+(log_8)(a)+(log_8)((b−4)/(c4))

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

(log_8)(4)+(log_8)(a)+(log_8)(b−4)−(log_8)(c4)

4. Apply the power rule for logarithms, which states that (log_b)(xp)=p*(log_b)(x) to the last term.

(log_8)(4)+(log_8)(a)+(log_8)(b−4)−4*(log_8)(c)

5. Simplify the constant term by evaluating (log_8)(4) Since 8=2 and 4=2 we have (log_2)(2)=2/3

2/3+(log_8)(a)+(log_8)(b−4)−4*(log_8)(c)

Final Answer

(log_8)((4*a*(b−4))/(c4))=2/3+(log_8)(a)+(log_8)(b−4)−4*(log_8)(c)

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