Problem 63 - Algebra 2

Question

Write true or false for each statement. Justify your answer. \(\frac{\log _{b} x}{\log _{b} y}=\log _{b} \frac{x}{y}\)

Step-by-Step Solution

Answer

The statement is false.

Step 1: Analyzing the Given Expression

Analyze the expression \(\frac{\log _{b} x}{\log _{b} y}\) and see if it can simplify to \(\log _{b} \frac{x}{y}\).

Step 2: Using the Change of Base Formula

Apply the change of base formula which states \(\log_{b}a = \frac{\log a}{\log b}\). Using this, rearrange the terms in the expression to give \(\frac{\log _{b} x}{\log _{b} y} = \frac{\log x /\log b}{\log y/\log b} = \frac{\log x}{\log y}\).

Step 3: Compare with the Right Hand Side

Ultimately, \(\frac{\log x}{\log y}\) is not equal to \(\log _{b} \frac{x}{y}\), so the statement is false.

Key Concepts

Change of Base Formula

Understanding logarithms can become more straightforward with the change of base formula. This technique allows you to rewrite a logarithm in terms of logarithms of a different base, often making the computations easier.

The formula states:

• \( \log_{b}(a) = \frac{\log_{k}(a)}{\log_{k}(b)} \)

Here, \(k\) can be any positive number and often, we choose it to be either 10 (common logarithm) or \(e\) (natural logarithm, denoted \(\ln\)).
Let's look at an example: if you want to find \(\log_{2}8\) using the change of base formula, using base 10, you can rewrite it as:

• \( \log_{2}8 = \frac{\log_{10}(8)}{\log_{10}(2)} \)

This conversion helps when using calculators, as many provide only \(\log_{10}\) or \(\ln\). This formula becomes especially handy when comparing different logarithmic expressions, such as in our original problem.

Logarithm Properties

Logarithms uphold several properties that allow you to manipulate and simplify expressions. These rules are handy for solving logarithmic equations and for rewriting complex expressions into simpler forms. Here are some key properties:

• Product Rule: \(\log_{b}(xy) = \log_{b}x + \log_{b}y\)
• Quotient Rule: \(\log_{b}\left(\frac{x}{y}\right) = \log_{b}x - \log_{b}y\)
• Power Rule: \(\log_{b}(x^n) = n\log_{b}x\)

In the context of our original exercise, it is important to think about the quotient rule and how it does not apply to the simplified form \(\frac{\log_{b} x}{\log_{b} y}\). Instead of simplifying to \(\log _{b} \frac{x}{y}\), it only provides the form \(\frac{\log x}{\log y}\) by using the change of base formula.

Expression Simplification

Expression simplification in logarithmic terms often involves converting one form to another to make calculations or comparisons easier.

In the original problem, the expression \(\frac{\log_{b} x}{\log_{b} y}\) needs simplifying to determine its correctness against \(\log_{b} \frac{x}{y}\). By applying the change of base formula, we discover that these two expressions are not equal.
For simplification, remember to:

• Utilize the change of base formula to re-express logarithms as ratios.
• Apply logarithmic properties judiciously to condense or expand expressions.

Through these maneuvers, the differences become evident, as the unchanged form doesn't simplify directly to its proposed counterpart. When simplifying, check all steps to ensure consistency with known rules and formulas. This accuracy is key to the simplification process.