Whenever a, b, and c are positive real numbers, which of the...

Whenever $a, b,$ and $c$ are positive real numbers, which of the following expressions is equivalent to $5 \log _{4} a-\log _{4} b+\frac{1}{3} \log _{2} c ?$ A. $5 \log _{4}\left(a^{5}-b\right)+\log _{2}\left(\frac{1}{3} c\right)$ B. $\log _{4} \frac{a^{5} b}{c}$ C. $\log _{4}\left(\frac{b}{a^{5}}\right)+\log _{2}\left(\frac{c}{3}\right)$ D. $5 \log _{4}\left(a^{5}-b\right)+\log _{2}\left(c^{3}\right)$ E. $\log _{4}\left(\frac{a^{5}}{b}\right)+\log _{2}(\sqrt[3]{c})$

Whenever $a, b,$ and $c$ are positive real numbers, which of the following expressions is equivalent to $5 \log _{4} a-\log _{4} b+\frac{1}{3} \log _{2} c ?$
A. $5 \log _{4}\left(a^{5}-b\right)+\log _{2}\left(\frac{1}{3} c\right)$
B. $\log _{4} \frac{a^{5} b}{c}$
C. $\log _{4}\left(\frac{b}{a^{5}}\right)+\log _{2}\left(\frac{c}{3}\right)$
D. $5 \log _{4}\left(a^{5}-b\right)+\log _{2}\left(c^{3}\right)$
E. $\log _{4}\left(\frac{a^{5}}{b}\right)+\log _{2}(\sqrt[3]{c})$

Key Concepts

Properties of Logarithms

This concept encompasses the basic rules governing the manipulation of logarithmic expressions, such as the product rule, quotient rule, and power rule. These rules allow one to simplify, combine, or break apart logarithms by converting operations like multiplication, division, and exponentiation into addition, subtraction, or multiplication of logarithms respectively.

Power Rule

The power rule of logarithms states that multiplying a logarithm by a factor is equivalent to raising the argument of the logarithm to that power. This is expressed as k·log_b(M) = log_b(M^k). It is particularly useful in problems where the logarithms involve coefficients that can be moved into exponents to simplify the expression.

Quotient and Product Rules

The quotient and product rules for logarithms allow the combination or separation of logarithmic terms when dealing with products or quotients. The product rule, log_b(MN) = log_b(M) + log_b(N), and the quotient rule, log_b(M/N) = log_b(M) - log_b(N), enable one to consolidate multiple logarithm terms into a single logarithm, or to decompose a single logarithm into multiple terms.

Change of Base Rule

The change of base rule allows logarithms with different bases to be converted into a common base, which is essential when simplifying or comparing logarithmic expressions. According to this rule, log_b(M) can be rewritten as log_k(M)/log_k(b) for any positive value of k. This helps in unifying logarithms with different bases into a form that is easier to handle and combine.

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