Problem 87 - A Graphical Approach to College Algebra

Question

Use the properties of logarithms to rewrite each expression as a single logarithm with coefficient 1. Assume that all variables represent positive real numbers. $$\ln (a+b)+\ln a-\frac{1}{2} \ln 4$$

Step-by-Step Solution

Answer

\( \ln(2a(a+b)) \)

Step 1: Apply the Product Rule of Logarithms

The product rule of logarithms states that \( \ln(x) + \ln(y) = \ln(xy) \). Apply this rule to the terms \( \ln(a+b) + \ln(a) \). This gives: \[ \ln((a+b) \cdot a) = \ln(a(a+b)) \].

Step 2: Apply the Power Rule of Logarithms

The power rule of logarithms is \( k \ln(x) = \ln(x^k) \). Apply this rule to \( -\frac{1}{2} \ln(4) \). This gives: \[ -\frac{1}{2} \ln(4) = \ln(4^{-1/2}) = \ln(\frac{1}{\sqrt{4}}) = \ln(\frac{1}{2}) \].

Step 3: Apply the Quotient Rule of Logarithms

The quotient rule of logarithms states that \( \ln(x) - \ln(y) = \ln\left(\frac{x}{y}\right) \). Apply this rule to combine the results from Step 1 and Step 2: \[ \ln(a(a+b)) - \ln(\frac{1}{2}) = \ln\left(\frac{a(a+b)}{\frac{1}{2}}\right) \].

Step 4: Simplify the Expression

Simplify the fraction inside the logarithm: \[ \frac{a(a+b)}{\frac{1}{2}} = a(a+b) \times 2 = 2a(a+b) \]. Therefore, the expression becomes \( \ln(2a(a+b)) \).

Key Concepts

Properties of Logarithms

Logarithms have unique properties that make them extremely useful in simplifying and manipulating expressions using logs. These properties include:

• Product Rule: Combines logarithms of multiplied values.
• Quotient Rule: Subtracts logs to divide corresponding values.
• Power Rule: Used to handle exponents inside the log.

Understanding these properties allows us to transform complex logarithmic expressions into simpler forms. This aids in both solving equations and understanding logarithmic behaviors in-depth. Each of these properties leverages the basic rules of exponents, which are inherent in the structure of logarithms themselves.

Product Rule of Logarithms

The product rule of logarithms is an essential tool when you come across the logarithm of a product. It states:

• If you have two numbers, say \(x\) and \(y\), their logarithms can be added as \(\ln(x) + \ln(y) = \ln(xy)\).
  
• This rule allows us to combine separate logs into a single logarithm.

In our exercise, \(\ln(a+b) + \ln(a)\) becomes \(\ln((a+b) \cdot a) = \ln(a(a+b))\).
By adding the logarithms of these two expressions using the product rule, we simplify the expression significantly, reducing two terms into one.

Power Rule of Logarithms

When dealing with a logarithm that has a coefficient in front, the power rule comes into play. This rule is described as:

• The expression \(k \ln(x)\) can be rewritten as \(\ln(x^k)\).
  
• This is particularly useful for turning multipliers in front of logs into an exponent on the number inside the log.

In the example from the exercise, the expression \(-\frac{1}{2} \ln(4)\) becomes \(\ln(4^{-1/2})\) or \(\ln\left(\frac{1}{\sqrt{4}}\right) = \ln\left(\frac{1}{2}\right)\).
Applying the power rule lets us remove the fraction from the front and incorporate it into the logarithm itself.

Quotient Rule of Logarithms

The quotient rule is used when you subtract one logarithm from another, and can be written as:

• \(\ln(x) - \ln(y) = \ln\left(\frac{x}{y}\right)\).
• This is useful for evaluating logs involving division, streamlining multiple log terms into a single term.

In the context of the exercise, after combining logs via the product rule and adjusting coefficients with the power rule, we use the quotient rule:
\(\ln(a(a+b)) - \ln\left(\frac{1}{2}\right)\), which simplifies to \(\ln\left(\frac{a(a+b)}{\frac{1}{2}}\right)\).
This step uses the rules of fractions, translating separate log terms into a clear division, simplified further by multiplying in the simplification step, arriving at \(\ln(2a(a+b))\).