Problem 86 - Intermediate Algebra

Question

Write each logarithmic expression as one logarithm. See Example 7. $$ \frac{1}{2} \log _{8}\left(x^{2}+5\right)-\log _{8}\left(x^{2}+5\right) $$

Step-by-Step Solution

Answer

\(\log_{8}(x^2 + 5)^{-1/2}\)

Step 1: Apply Logarithm Power Rule

The first term in the expression involves the logarithm power rule, which states that \(a \log_b (M) = \log_b (M^a)\). Therefore, \(\frac{1}{2} \log_{8}(x^{2}+5)\) becomes \(\log_{8}((x^2+5)^{1/2})\).

Step 2: Apply Logarithm Subtraction Rule

The expression \(\log_{8}((x^2+5)^{1/2}) - \log_{8}(x^2+5)\) can be combined using the rule for subtracting logarithms: \(\log_b (M) - \log_b (N) = \log_b \left(\frac{M}{N}\right)\). Substituting the values, the expression becomes \(\log_{8}\left(\frac{(x^2+5)^{1/2}}{x^2+5}\right)\).

Step 3: Simplify the Fraction Inside the Logarithm

Simplify the fraction \(\frac{(x^2+5)^{1/2}}{x^2+5}\). Notice that \((x^2+5)^{1/2}\) is the square root of \(x^2+5\), so it simplifies to \(\frac{\sqrt{x^2+5}}{x^2+5}\). Therefore, the original logarithmic expression becomes \(\log_{8}\left(\frac{1}{\sqrt{x^2+5}}\right)\).

Step 4: Express as a Single Logarithm

The expression \(\log_{8}\left(\frac{\sqrt{x^2+5}}{x^2+5}\right)\) simplifies within the logarithm to \(\log_{8}\left((x^2+5)^{-1/2}\right)\), using the property that \(\frac{1}{a} = a^{-1}\).

Key Concepts

Understanding Logarithms

Logarithms are a way to express powers or exponents. Imagine you're thinking of a number that, when multiplied a certain number of times, becomes another number. That's basically what a logarithm is doing in reverse.
For example, the logarithm to base 10 of 100 is 2, because 10 squared (10 times 10) is 100.

• The base of the logarithm tells us what number is being repeatedly multiplied.
• The answer to a logarithm tells us how many times the base has been multiplied.

Most logarithms you'll encounter in algebra have specific rules that make them easier to handle. For instance, you can learn about the power rule, which lets us move exponents in and out of the logarithm, and the subtraction (quotient) rule, which lets us simplify expressions like the ones in the given problem.

Basic Concepts of Algebra

Algebra is like a language in mathematics where we use letters to represent unknown numbers or variables. This lets us create formulas and solve lots of real-world problems.
When working with expressions in algebra, you might encounter terms that include numbers, variables, and sometimes exponents. In the context of logarithms, these can be expressed using specific rules.

• Variables can stand in for any number, and when combined with operations like addition or multiplication, they form expressions.
• The rules of algebra, like distributing and combining like terms, make it possible to simplify complex expressions.

In our logarithmic expression, understanding these basic algebraic rules helps us transform and simplify expressions, making them easier to work with.

Simplifying Expressions

Simplifying expressions is about making them as concise as possible without changing their value. It's a key skill in mathematics that reduces complexity and emphasizes clarity.
To simplify logarithmic expressions, we often use certain rules, like the power rule or the subtraction (quotient) rule:

• Power Rule: This rule lets you pull an exponent out of the logarithm. For instance, \( a \log_b (M) = \log_b (M^a) \).
• Subtraction (Quotient) Rule: This allows you to turn a subtraction into division inside the log. Like \( \log_b (M) - \log_b (N) = \log_b \left(\frac{M}{N}\right) \).

In the original exercise, simplifying meant applying these rules step-by-step to express a complex logarithmic statement into a cleaner, single logarithm format. Each step reduces unnecessary complexity, bringing focus to the main components of the expression.