Problem 82 - Elementary and Intermediate Algebra

Question

Write logarithmic expression as one logarithm. \(\frac{1}{4}\left[\log _{r}\left(n^{2}-16\right)-\log _{r}(n-4)\right]\)

Step-by-Step Solution

Answer

The expression is \(\log_{r}((n+4)^{1/4})\).

Step 1: Understanding the Expression

We need to simplify the expression \(\frac{1}{4}\left[\log_{r}(n^2-16)-\log_{r}(n-4)\right]\) by combining the logarithms into one. We first focus on the difference in the logarithms.

Step 2: Combining the Logarithms

Using the property \(\log_{b}(A) - \log_{b}(B) = \log_{b}\left(\frac{A}{B}\right)\), we can rewrite \(\log_{r}(n^2-16)-\log_{r}(n-4)\) as \(\log_{r}\left(\frac{n^2-16}{n-4}\right)\).

Step 3: Simplifying the Fraction

Notice that \(n^2 - 16\) is a difference of squares, which factors as \((n-4)(n+4)\). Therefore, \(\frac{n^2-16}{n-4}\) simplifies to \(n+4\) for \(n eq 4\).

Step 4: Rewriting the Expression

Now, the expression becomes \(\frac{1}{4}\log_{r}(n+4)\).

Step 5: Incorporating the Constant Factor

Using the property \(c \cdot \log_{b}(A) = \log_{b}(A^c)\), we rewrite \(\frac{1}{4}\log_{r}(n+4)\) as \(\log_{r}((n+4)^{1/4})\).

Step 6: Final Expression

The original expression is rewritten as one logarithm: \(\log_{r}((n+4)^{1/4})\).

Key Concepts

Logarithm Properties

Understanding the properties of logarithms is key to solving problems involving them. Logarithms are mathematical functions that serve as the inverse of exponential functions. Several properties help simplify expressions and solve equations more effectively.

• Difference of Logarithms: If you have two logarithms with the same base subtracted from one another, such as \(\log_{b}(A) - \log_{b}(B)\), this can be rewritten as a single logarithm: \(\log_{b}\left(\frac{A}{B}\right)\). This property allows you to consolidate expressions into a more manageable form.
• Constant Multiplication: When a logarithm is multiplied by a constant, for instance \(c \cdot \log_{b}(A)\), it can be converted into \(\log_{b}(A^c)\). This means raising the expression inside the logarithm to the power of the constant.

These properties are essential for simplifying and solving logarithmic expressions, as demonstrated in the original exercise where we consolidated separate logs into one.

Difference of Squares

The difference of squares is a useful concept when simplifying expressions, particularly with polynomials inside logarithms. It's a specific algebraic identity: \(a^2 - b^2 = (a-b)(a+b)\).
In the exercise provided, the concept is applied as follows:

• The expression \(n^2 - 16\) is recognized as a difference of squares because it matches the form \(a^2 - b^2\), where \(a = n\) and \(b = 4\).
• Simplifying this gives us \((n-4)(n+4)\), aiding in reducing more complex expressions.

This factorization of the original expression allows the cancellation of terms, essential for simplifying fractions and combining logarithms as one. Recognizing the difference of squares can be a powerful shortcut in algebra, making it simpler to work through complex mathematical problems.

Simplifying Logarithms

Once you understand the basic properties of logarithms and factorization techniques like the difference of squares, the next step is simplifying logarithmic expressions. This often involves a series of substitutions and reductive steps.

• Take our previous steps: After using the property \(\log_b(A) - \log_b(B) = \log_b\left(\frac{A}{B}\right)\), we simplified the fraction by recognizing and utilizing the difference of squares. This led to \(\log_{r}(n+4)\).
• The final step in our original exercise involved incorporating a constant factor. We employed the property that allows us to express \(\frac{1}{4}\log_{r}(n+4)\) as \(\log_{r}((n+4)^{1/4})\). This transforms our logarithm into an even simpler expression.

Simplifying logarithms involves utilizing these properties strategically to handle complex expressions, like the ones encountered in our exercise. Mastering these steps enables easy manipulation of logarithmic expressions in advanced mathematical problems.