Problem 48 - College Algebra and Calculus: An Applied Approach

Question

In Exercises, write the expression as the logarithm of a single quantity. $$ 2\left[\ln x+\frac{1}{4} \ln (x+1)\right] $$

Step-by-Step Solution

Answer

The simplified single logarithmic function is \( \ln \left[x^2 (x+1)^{1/2}\right] \)

Step 1: Applying the power property of logarithm

Multiplying the two logarithmic functions by 2, results in: \( 2\ln x+ \frac{1}{2} \ln (x+1) \).

Step 2: Simplifying the expression

Applying the power property of logarithm, \(a \log_b m\) can be rewritten as \( \log_b m^a \): \( \ln x^2+ \ln (x+1)^{\frac{1}{2}} \).

Step 3: Combining Logarithms

Applying the product rule of logarithms, \( \log_b A + \log_b B = \log_b AB \), results in: \( \ln \left[x^2 (x+1)^{1/2}\right] \).

Key Concepts

Logarithm Properties

Logarithms are an essential tool in mathematics, especially when it comes to dealing with exponential functions. They have unique properties that allow us to manipulate and combine logarithmic expressions efficiently. Understanding these properties is crucial for both simple calculations and more complex algebraic manipulations.

Firstly, the logarithm property that states \[ \log_b (mn) = \log_b m + \log_b n \] demonstrates how the logarithm of a product can be expressed as the sum of logarithms. Similarly, the quotient rule \[ \log_b \left( \frac{m}{n} \right) = \log_b m - \log_b n \] shows how the logarithm of a quotient is the difference between the logarithms. Another important property is that \[ \log_b b = 1 \] and \[ \log_b 1 = 0 \], which are useful for simplifying equations.

When solving a problem involving logarithms, identifying which properties can be applied helps simplify expressions and can make solving for variables more straightforward. For example, in the provided exercise, we would initially recognize the opportunity to apply both the power property and the product rule of logarithms to simplify the expression.

Power Property of Logarithm

One of the powers of logarithms is, quite literally, the power property. This property allows us to take an exponent and move it to the front of the logarithm, turning multiplication into addition or division, which are often easier to work with.

The power property states that for any real number \(a\) and positive real numbers \(b\) and \(m\), \[ a \log_b m = \log_b m^a \]. This means if you have a logarithm being multiplied by a coefficient, you can rewrite it as a single logarithm with the argument raised to the power of that coefficient. This feature is incredibly handy when you want to combine multiple logarithmic terms into one.

For instance, the textbook exercise began with the expression \( 2\ln x+ \frac{1}{2} \ln (x+1) \). By applying the power property, the terms become \( \ln x^2+ \ln (x+1)^{\frac{1}{2}} \) before further simplification.

Product Rule of Logarithms

Once individual logarithmic terms are simplified using properties like the power property, we can combine them into a single expression using the product rule of logarithms. This property states that the sum of two logarithms with the same base can be combined into the logarithm of the product of their arguments, summarized as \[ \log_b (A) + \log_b (B) = \log_b (AB) \].

This rule is incredibly useful when trying to condense an expression or solve equations that feature logarithmic terms. In our exercise, after utilizing the power property, the product rule is applied to combine \( \ln x^2 \) and \( \ln (x+1)^{\frac{1}{2}} \) to obtain \( \ln \left[x^2 (x+1)^{1/2}\right] \). This final expression is much simpler and represents the logarithm of a single quantity as required by the original problem.

Taking the steps to first apply the power property, and then the product rule, is a systematic approach to simplify and solve logarithmic expressions effectively.