Question

Use the properties of logarithms to expand the expression as a sum, difference, and/or multiple of logarithms. (Assume all variables are positive.)\(\log _{9} \frac{\sqrt{y}}{z^{2}}\)

Short answer

The expanded form of the given log expression is \(\frac{1}{2} \log _{9} y - 2 \log _{9} z\).

Step-by-step solution

Convert the Logarithmic Expression to a Difference

Applying the first log rule, we can express the fraction inside the logarithm as a difference of two logarithms. Thus, \(\log _{9} \frac{\sqrt{y}}{z^{2}}\) becomes \(\log _{9} \sqrt{y} - \log _{9} {z^{2}}\).

Apply the Second Property to Express the Root and Power as Multiples

Next, use the second log rule to simplify the square root and the square power in the terms. Hence, the expression becomes: \(\frac{1}{2} \log _{9} y - 2 \log _{9} z\).

Key concepts

Logarithmic Expressions

Logarithmic expressions represent the power to which a base number must be raised to obtain a certain value. Understanding these expressions is crucial, as they help to solve equations where the unknown variable is an exponent.

In the given exercise, we start with the logarithmic expression \( \log_{9} \frac{\sqrt{y}}{z^{2}} \). This expression asks, 'To what power must 9 be raised to equal the fraction \( \frac{\sqrt{y}}{z^{2}} \)?' Logarithms turn the process of exponentiation on its head, allowing us to work with the powers instead of the numbers themselves. This makes solving complex algebraic problems more manageable.

When approaching logarithmic expressions, we seek to decompose them to their simplest forms using various properties and rules of logarithms, making it easier to manage and solve the equations they represent.

Expanding Logarithms

Expanding logarithms is the process of breaking down a single logarithmic expression into multiple terms that are easier to work with. This is effectively the reverse of condensing logarithms, which combines multiple logarithmic terms into one.

The provided exercise is an example of expanding a logarithm. It begins with a combined logarithmic expression and proceeds to split it into separate, simpler logarithmic terms. By expanding \( \log_{9} \frac{\sqrt{y}}{z^{2}} \), we turn it into two terms involving the square root of \( y \) and the square of \( z \) separately.

Understanding this process is invaluable, as it lays the groundwork for further manipulation in solving logarithmic equations. By mastering expansion, students gain the ability to rearrange equations into a form that is easier for them to solve, calculate derivatives in calculus, and tackle problems in exponential growth and decay.

Logarithm Rules

There are several key rules or properties of logarithms that are essential tools in simplifying logarithmic expressions. The steps in the exercise demonstrate two of these rules:

• The Quotient Rule: \( \log_{b}(\frac{x}{y}) = \log_{b}(x) - \log_{b}(y) \), which allows us to express the logarithm of a quotient as the difference of logarithms.
• The Power Rule: \( \log_{b}(x^{n}) = n \cdot \log_{b}(x) \), which permits us to move the exponent in a logarithmic argument out front as a multiplier.

These rules help us to decompose complex logarithmic expressions into simpler forms. The Quotient Rule is applied first in the solution to split the original logarithm into two parts, and the Power Rule further simplifies each part by addressing the exponent within the argument of each resulting logarithm.

Logarithm Properties

The Base and Argument

Logarithms are not just about rules. Their properties are grounded in the relationship between the base and the argument. The base, in this case 9, decides the 'language' in which the logarithmic 'question' is asked. The argument, which is the value or expression inside the logarithm, is what we're attempting to translate into this logarithmic language.

Other properties include:

• Logarithms of products and powers allow for multiplication and exponents to be broken down into addition and multiplication of logarithms, as demonstrated in the exercise.
• The Change of Base Formula, a sneaky trick that allows us to recalculate logs in terms of a base more convenient to us, often base 10 or the natural base \( e \).
• And importantly, the understanding that \( \log_{b}(1) = 0 \) and \( \log_{b}(b) = 1 \), which reflect the zero power and the fundamental property of exponents, respectively.

Having a solid grasp of these properties empowers students to navigate comfortably through logarithmic calculations.