Express each of the following as the sum or difference of...

Express each of the following as the sum or difference of simpler logarithmic quantities. Assume all variables represent positive real numbers. (Objective 4$)$ For example, $$\begin{aligned} \log _{b} \frac{x^{3}}{y^{2}} &=\log _{b} x^{3}-\log _{b} y^{2} \\ &=3 \log _{b} x-2 \log _{b} y \end{aligned}$$ $$ \log _{b} \sqrt{\frac{x}{y}} $$

Express each of the following as the sum or difference of simpler logarithmic quantities. Assume all variables represent positive real numbers. (Objective 4$)$ For example,
$$\begin{aligned}
\log _{b} \frac{x^{3}}{y^{2}} &=\log _{b} x^{3}-\log _{b} y^{2} \\
&=3 \log _{b} x-2 \log _{b} y
\end{aligned}$$
$$
\log _{b} \sqrt{\frac{x}{y}}
$$

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Key Concepts

Logarithmic Quotient Rule

This rule states that the logarithm of a quotient can be expressed as the difference of two logarithms. In general, for any positive real numbers a, b, and c, the logarithmic expression log_b(a/c) is equivalent to log_b(a) - log_b(c). This property is fundamental in breaking down complex logarithmic expressions into simpler parts.

Logarithmic Power Rule

The power rule for logarithms allows you to bring an exponent from the argument of the logarithm to the front as a multiplier. That is, for a positive real number a and any real number r, log_b(a^r) is equal to r times log_b(a). This technique is essential for simplifying expressions where variables are raised to a power.

Conversion of Radicals to Exponents

Radicals can be rewritten as fractional exponents, which makes them easier to work with when applying logarithmic properties. For instance, the square root of an expression x is written as x^(1/2). Converting radicals to exponent form allows the use of the logarithmic power rule to simplify the expression further.

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