Question

Find a logarithmic equation that relates \(y\) and \(x\). Explain the steps used to find the equation.Reasoning An algebra student claims that the following is true: \(\log _{a} \frac{x}{y}=\frac{\log _{a} x}{\log _{a} y}=\log _{a} x-\log _{a} y\) Discuss how you would use a graphing utility to demonstrate that this claim is not true. Describe how to demonstrate the actual property of logarithms that is hidden in this faulty claim.

Short answer

The student's claim is partially correct. While it is true that \(\log _{a}(x/y) = \log _{a}(x) - \log _{a}(y)\) according to the properties of logarithms, the assertion that \(\log _{a}(x/y) = \frac{\log _{a}(x)}{\log _{a}(y)}\) is incorrect. This can be demonstrated by graphing both function types and seeing that they don't yield the same result.

Step-by-step solution

Understanding the Student's Claim

The student is making two claims here. The first claim is that \(\log _{a}{(x/y)}\) is equivalent to \(\frac{\log _{a}(x)}{\log _{a}(y)}\), and the second claim is that \(\log _{a}{(x/y)}\) is equivalent to \(\log _{a}(x) - \log _{a}(y)\). While the second equivalence actually holds true (according to the properties of logarithms), the first one does not. This will become clear in the next steps.

Demonstrating Incorrectness of the First Claim

To show that the first equality doesn't hold, you would plot two functions using a graphing utility: \( f(x) = \log _{a}{x/y} \) (for a specific choice of \(a\), \(y\), different from \(x\)) and \( g(x) = \frac{\log _{a}(x)}{\log _{a}(y)} \). If the graphs of both functions don't coincide for all allowable values of \(x\), then the two expressions are not equivalent. In this case, they will indeed not coincide, proving that the student's first claim is incorrect.

Demonstrating Correctness of the Second Part of the Claim

In contrast to the first claim, the second part is correct according to the properties of logarithms. The quotient rule states that the logarithm of a quotient is equal to the difference of the logarithms. This means that \(\log _{a}(x/y) = \log _{a}(x) - \log _{a}(y) \) for \(x, y > 0\). To demonstrate this, you could plot the two functions \( h(x) = \log _{a}{x/y} \) and \( j(x) = \log _{a}(x) - \log _{a}(y) \) using a graphing utility. In this case, you would observe that the graphs coincide, thus verifying the second part of the student's claim.

Key concepts

Properties of Logarithms

The properties of logarithms help us simplify complex expressions and solve logarithmic equations efficiently. Understanding these properties is crucial for dealing with logarithmic expressions like the one in this exercise. Let's break down the key properties you should know:

• Product Rule: This rule states that \( \log_a(xy) = \log_a x + \log_a y \). It allows us to break down a single logarithm of a product into a sum of two logarithms.


• Quotient Rule: According to this rule, \( \log_a(x/y) = \log_a x - \log_a y \). This is the accurate property that supports the student's correct claim and allows us to handle divisions within logarithms.


• Power Rule: If you have an expression such as \( \log_a(x^b) \), this will be equal to \( b \cdot \log_a x \). This lets us deal with exponents efficiently within logarithmic expressions.

In essence, the properties of logarithms serve as powerful tools to transform and solve different sets of logarithmic problems.

Graphing Utilities

Graphing utilities like graphing calculators or software are excellent tools to visually verify or refute mathematical claims. In our scenario with logarithmic equations, a graphing utility can easily demonstrate whether two expressions are equivalent. Here's how you can use one effectively:

• Input the first logarithmic function, such as \( f(x) = \log_a(x/y) \).


• Then input the second function, \( g(x) = \frac{\log_a(x)}{\log_a(y)} \).


• Compare the resulting graphs. If they don't match for all allowed values, the expressions are not equivalent.

The graphing utility can also confirm correct claims by displaying coinciding graphs when two expressions are indeed equal, such as \( \log_a(x/y) \) and \( \log_a(x) - \log_a(y) \). This visual representation helps solidify understanding and dispels any doubts surrounding equivalences of logarithmic equations.

Logarithm Rules

Logarithm rules are fundamental guidelines that dictate how we manipulate and interact with logarithmic expressions. The rules are not only mathematically sound but also quite intuitive once understood clearly.

• Logarithmic Identity: The rule \( \log_a(a) = 1 \) is simple but fundamental. It tells us that any logarithm with the same base and argument equals one.


• Change of Base Formula: This is useful when calculator limitations force you to convert logarithms. It is expressed as \( \log_b(x) = \frac{\log_k(x)}{\log_k(b)} \), where \(k\) is a new base like 10 or \(e\).


• Logarithm of 1: The rule \( \log_a(1) = 0 \) is intuitive, stating that any number raised to the power of zero will equal 1.

Understanding these rules allows anyone to tackle logarithmic problems with confidence and clarity. They are the foundation of solving equations and comparing different logarithmic functions.