Problem 67 - College Algebra

Question

In Exercises \(41-70,\) use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(1 .\) Where possible, evaluate logarithmic expressions without using a calculator. $$ \frac{1}{3}\left[2 \ln (x+5)-\ln x-\ln \left(x^{2}-4\right)\right] $$

Step-by-Step Solution

Answer

\(\ln[\frac{(x+5)^2}{x^3-4x}]^{1/3} \)

Step 1: Apply the power rule of logarithms

Use the power rule of logarithms to move the coefficients inside the logarithms as powers. The power rule states that \(a \log b = \log (b^a)\). Therefore, the equation becomes: \( \frac{1}{3}[\ln ((x+5)^2)-\ln x-\ln ((x^2-4))]\).

Step 2: Combine into a single logarithm

Use the properties of logarithms to combine the logarithms into one. The relevant properties are as follows: For any positive numbers m, n, and r, where \(m \neq 1\) and \(n \neq 1\) \(\log_m nr = \log_m n + log_m r\) and \(\log_m (n/r) = \log_m n - \log_m r\). Hence, we get: \( \frac{1}{3}[\ln \frac{(x+5)^2}{x(x^2-4)}]\).

Step 3: Simplify

Simplify the expression inside the logarithm. This results in: \( \frac{1}{3}[\ln \frac{(x+5)^2}{x^3-4x}]\).

Step 4: Apply coefficient outside the log as power inside

Apply the power rule of logarithms to move the coefficient inside the log as power. Hence, the simplified expression is: \( \ln[\frac{(x+5)^2}{x^3-4x}]^{1/3} \).

Key Concepts

Condensing Logarithms

When working with logarithms, it's often necessary to combine multiple logarithmic expressions into a single term; this process is known as condensing logarithms. Condensing is particularly useful for simplification and solving equations. The objective is to write the expression as a single logarithm with a coefficient of 1.

How is this done? Typically, it involves using the properties of logarithms, such as the product, quotient, and power rules. For instance, suppose you have the terms ) and ). These can be condensed into one term by using the quotient property: ).) can be expressed as a single logarithm, illustrating the essence of condensing logarithms.

Condensing not only tidies up the expression but also prepares it for further operations, such as differentiation or integration in calculus, or for evaluating the expression numerically if need be.

Power Rule of Logarithms

The power rule of logarithms is a pivotal property that enables us to deal with exponents within logarithmic functions. In its basic form, the power rule can be expressed as follows: if you have an expression like ), you can rewrite it as ). Essentially, the power rule allows us to move the coefficient in front of a log to the exponent on the argument of the log.

In our example, ) is employed to transform the coefficients into powers of the arguments, resulting in ).) after applying this property. This manipulation is crucial for condensing logarithms as it structure of logarithmic expressions to make things simpler.

Combining Logarithms

The art of combining logarithms lies in using properties to merge several logarithmic terms into a singular expression. This is useful in equations where we have several logs with the same base that we want to merge for the purpose of simplification or solving.

Consider the properties that allow us to combine logs: The product rule states that ), while the quotient rule tells us that ). Employing these rules, we can combine the terms in our example: from ) and ) we get ).^{1/3}). In this way, we streamline multiple log terms into one neater expression.