Problem 111 - Algebra and Trigonometry

Question

You overhear a student talking about a property of logarithms in which division becomes subtraction. Explain what the student means by this.

Step-by-Step Solution

Answer

The property of logarithms in question is the Quotient Rule of Logarithms, which states that the logarithm of a quotient is the difference of the logarithms. It means division inside the log can be translated to subtraction outside the log, such as \( \log_a\left(\frac{x}{y}\right) = \log_a(x) - \log_a(y)\).

Step 1: Introduction to Logarithmic Property

The logarithmic property in question is often referred to as the Quotient Rule of Logarithms. It is one of the three basic properties of logarithms, together with the Product Rule and the Power Rule, which allow you to simplify expressions or solve equations involving logarithms.

Step 2: The Quotient Rule of Logarithms Explained

The Quotient Rule states that the logarithm of a quotient is the difference of the logarithms. In mathematical terms, it can be expressed as follows: If you have two positive real numbers, \(a\) and \(b\), and \(a \neq 1\), then for every \(x > 0\) and \(y > 0\), the rule is as such: \( \log_a\left(\frac{x}{y}\right) = \log_a(x) - \log_a(y)\) . This means, when you are dividing within a log expression, you can convert it to subtraction outside of the log.

Step 3: Example

For example, suppose we have \( \log_2\left(\frac{8}{4}\right)\). Using the Quotient Rule of Logarithms, this division inside the logarithm can be rewritten as subtraction outside the logarithm, like so: \( \log_2\left(\frac{8}{4}\right) = \log_2(8) - \log_2(4)\). After calculating the log values, it becomes \(3 - 2 = 1\), which is indeed the same as \( \log_2(2)\).