Problem 71 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the properties of logarithms to condense the expression.$$\log _{4} z-\log _{4} y$$.

Step-by-Step Solution

Answer

The condensed form of the expression \(\log_{4} z - \log_{4} y\) is \(\log_{4} \left(\frac{z}{y}\right)\).

Step 1: Recognize the properties of logarithms

The properties of logarithms allow us to rewrite expressions in different forms. In this case, we need the quotient rule, which states: \(\log_b(a) - \log_b(b) = \log_b\left(\frac{a}{b}\right)\). Here, \(a\) and \(b\) are positive real numbers, and \(b != 1\).

Step 2: Apply the quotient rule

Applying the quotient rule to our expression, it rewrites the difference of two logarithms with the same base, \(\log_b(a) - \log_b(b)\), as a single logarithm of the quotient of the values: \(\log_{4} z - \log_{4} y = \log_{4} \left(\frac{z}{y}\right)\).

Key Concepts

Quotient Rule

The quotient rule is a fundamental property of logarithms that simplifies the subtraction of logarithmic expressions. This rule states that when you have the difference of two logarithms with the same base, the result can be condensed into a single logarithm of a quotient. Here’s how it works:

• Consider \(\log_b(a) - \log_b(b)\). These are two logarithmic terms with the same base, \(b\).
• According to the quotient rule, this expression can be rewritten as \(\log_b\left(\frac{a}{b}\right)\).

This means we turn a subtraction problem into a division problem within a single logarithm.
In the given exercise, the expression \(\log _{4} z - \log _{4} y\) becomes \(\log_4\left(\frac{z}{y}\right)\). This demonstrates how the quotient rule helps in simplifying logarithmic expressions efficiently. Understanding this rule makes working with logarithms much easier and clearer, especially in solving equations and manipulating expressions.

Logarithmic Expressions

Logarithmic expressions are mathematical statements that involve logarithms, which are the inverses of exponential functions. A logarithm answers the question: "To what power must the base be raised to yield a certain number?"
For example, in \(\log_b(a)\), \(b\) is the base, \(a\) is the argument of the logarithm, and the equation is looking for the power (or logarithm) that results in \(a\).
Logarithms are useful when dealing with exponential growths and can often condense or expand mathematical equations:

• Condensing uses properties, like the quotient rule, to simplify expressions.
• Expanding can make expressions easier to compute manually by breaking them into smaller parts.

These expressions appear in many fields such as science, engineering, and finance, assisting in understanding phenomena that involve large scales or ranges.

Condense Expressions

Condensing logarithmic expressions is a process used to combine multiple logarithms into a simpler, single logarithmic term using the properties of logarithms.
This helps to:

• Simplify complex mathematical statements, making them easier to work with.
• Streamline calculations in both algebraic computations and real-world applications.

When condensing, we employ rules such as:

• The quotient rule, to handle subtractions, turning them into divisions inside a logarithm.
• The product rule, to manage additions, transforming them into multiplications inside the logarithm.
• The power rule, which helps when dealing with coefficients by converting them into exponents.

In our exercise, the expression \(\log _{4} z - \log _{4} y\) is condensed into \(\log_{4}\left(\frac{z}{y}\right)\) by using the quotient rule. This showcases how condensing turns a more complex expression into a simpler form, making it much easier to interpret and solve when incorporated into larger mathematical problems.