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 ((x²y)/z²)

Verified step by step guidance

1. Recall the logarithm property for division: $\log b_{}\left(mn\right)=\log b_{}\left(m\right)-\log b_{}\left(n\right)$. Apply this to separate the logarithm of the fraction into a difference of two logarithms.
2. Apply the logarithm property for multiplication: $\log b_{}(m\cdot n)=\log b_{}\left(m\right)+\log b_{}\left(n\right)$. Use this to expand the logarithm of the numerator $x^2\; y$ into a sum of logarithms.
3. Use the power rule of logarithms: $\log b_{}\left(a^{\mathrm{n}}\right)=\mathrm{n}\log b_{}\left(a\right)$. Apply this to the terms $x^2$ and $z^2$ to bring the exponents out in front of the logarithms.
4. Combine all the expanded parts: write the expression as the sum and difference of logarithms with coefficients, specifically $2\; \backslash ,\; log\_b(x)\; +\; log\_b(y)\; -\; 2\; \backslash ,\; log\_b(z)$.
5. Check if any of the logarithmic expressions can be simplified further or evaluated without a calculator, depending on the values of $x$, $y$, and $z$. If no values are given, the expansion is complete.

Key Concepts

Properties of Logarithms

Properties of logarithms include the product, quotient, and power rules, which allow the expansion or simplification of logarithmic expressions. For example, log_b(MN) = log_b(M) + log_b(N), log_b(M/N) = log_b(M) - log_b(N), and log_b(M^k) = k * log_b(M). These rules help break down complex expressions into simpler parts.

Logarithmic Expansion

Logarithmic expansion involves rewriting a logarithm of a product, quotient, or power as a sum, difference, or multiple of logarithms. This process makes it easier to simplify or evaluate expressions, especially when variables and exponents are involved, as in log_b((x^2 y)/z^2).

Evaluating Logarithms Without a Calculator

Evaluating logarithms without a calculator requires recognizing when expressions can be simplified using known logarithmic values or properties. For example, if variables represent known values or if the expression reduces to log_b(b^k), it can be evaluated directly as k. This skill is essential for exact answers in algebra.