Question

Condense the expression to the logarithm of a single quantity.\(\log _{8}(x-2)-\log _{8}(x+2)\)

Short answer

\(\log_{8}\frac{x-2}{x+2}\)

Step-by-step solution

Identify and apply the logarithmic quotient rule

According to the properties of logarithms, the difference of two logarithms (with same base) equals the logarithm of the fraction of their quantities. Therefore, the expression \(\log_{8}(x-2)-\log_{8}(x+2)\) can be written as \(\log_{8}\frac{x-2}{x+2}\).

Simplification

At this point, there are no further steps for simplification. Therefore, \(\log_{8}\frac{x-2}{x+2}\) is the simplified version of the expression to the logarithm of a single quantity.

Key concepts

Logarithm Properties

To understand the original exercise, it's vital to grasp the key properties of logarithms. Logarithms have a special set of rules, or properties, that help us manipulate and simplify logarithmic expressions.

• Product Rule: This rule states that the logarithm of a product equals the sum of the logarithms of the individual factors. Mathematically, this is expressed as \( \log_b(M \cdot N) = \log_b(M) + \log_b(N) \).
• Quotient Rule: The quotient rule allows us to express the logarithm of a quotient as the difference of two logarithms. It can be described as \( \log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N) \).
• Power Rule: This rule applies when you have an exponent inside a logarithm. It states that \( \log_b(M^n) = n \cdot \log_b(M) \).

These properties are incredibly useful in many mathematical contexts, especially in simplifying and solving logarithmic expressions, just like in the given exercise above.

Logarithmic Expressions

A logarithmic expression is a mathematical expression that involves the logarithm of a number or variable. In the context of the original exercise, we dealt with the expression \( \log_{8}(x-2) - \log_{8}(x+2) \).
Logarithmic expressions often involve mathematical operations like addition, subtraction, multiplication, division, and exponents. All these operations can be simplified using the properties of logarithms, making expressions easier to evaluate or analyze.
Understanding how to manipulate and simplify logarithmic expressions enables us to solve equations and model real-world scenarios where exponential relationships occur, such as in finance or science. The base of logarithms, which in the problem is 8, provides the context for these expressions, indicating how many times a base number is multiplied to achieve a given quantity.
Recognizing these expressions and knowing how to transform them correctly is essential for advancing to more challenging logarithmic problems.

Simplifying Logarithms

Simplifying logarithms is an essential skill when working with logarithmic functions. The process often involves applying the properties of logarithms to reduce complex expressions into simpler forms.
In the exercise, the expression \( \log_{8}(x-2) - \log_{8}(x+2) \) was condensed into a single logarithm using the quotient rule. We applied this rule because the expression was a difference of logarithms, making it suitable for conversion into the logarithm of a fraction.
The result, \( \log_{8}\left(\frac{x-2}{x+2}\right) \), is much more streamlined and less complicated to work with. Simplifying logarithms doesn't stop with just using one property; depending on the scenario, multiple properties may be used in conjunction to achieve the least complex form.
These practices are not only useful for problem-solving but also for gaining deeper insights into how logarithmic functions behave and interact with other mathematical concepts.