In Exercises 1–40, use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. log_b ((√x y³)/z³)

Verified step by step guidance

1. Recall the logarithm property for a quotient: $\log b_{}\left(\frac{A}{B}\right)=\log b_{}\left(A\right)-\log b_{}\left(B\right)$. Apply this to separate the logarithm of the fraction into a difference of logarithms.
2. Apply the logarithm property for a product: $\log b_{}\left(AC\right)=\log b_{}\left(A\right)+\log b_{}\left(C\right)$. Use this to expand the logarithm of the product in the numerator, $\sqrt{x}y^3$, into a sum of logarithms.
3. Rewrite the square root and powers inside the logarithms using the power rule: $\log b_{}\left(\sqrt{x}\right)=\log b_{}\left(x\right)^\{\frac{1}{2}\}$ and $\log b_{}\left(y\right){)}^3=\; 3\log b_{}\left(y\right)$. Similarly, apply the power rule to the denominator term $z^3$.
4. Express each logarithm with the exponent brought in front as a coefficient: $a\log b_{}\left(m\right)$, where $a$ is the exponent.
5. Combine all the expanded terms into a single expression: the sum of the logarithms from the numerator minus the logarithm from the denominator, each with their respective coefficients.

Key Concepts

Properties of Logarithms

Properties of logarithms include rules such as the product rule, quotient rule, and power rule. These allow you to expand or condense logarithmic expressions by turning products into sums, quotients into differences, and exponents into multipliers, facilitating simplification and evaluation.

Radicals and Exponents

Understanding how to express radicals as fractional exponents is essential. For example, the square root of x can be written as x^(1/2), which allows the use of logarithm power rules to simplify expressions involving roots.

Logarithmic Expression Simplification

Simplifying logarithmic expressions involves applying the properties systematically to break down complex expressions into sums and differences of simpler logarithms. This process often reveals terms that can be evaluated or further simplified without a calculator.