Problem 30 - Algebra for College Students

Question

Solve each logarithmic equation and express irrational solutions in lowest radical form. $$ \ln (3 t-4)-\ln (t+1)=\ln 2 $$

Step-by-Step Solution

Answer

The solution is \( t = 6 \).

Step 1: Apply Logarithmic Properties

Apply the property of logarithms that states \( \ln a - \ln b = \ln\left(\frac{a}{b}\right) \). This gives us the equation: \[ \ln \left( \frac{3t-4}{t+1} \right) = \ln 2. \]

Step 2: Eliminate the Logarithms

Since the natural logarithm on both sides of the equation is equal, we can set the expressions inside the logarithms equal to each other. Thus: \[ \frac{3t-4}{t+1} = 2. \]

Step 3: Solve the Rational Equation

Cross multiply to solve the equation: \[ 3t - 4 = 2(t + 1). \] Simplify the equation: \[ 3t - 4 = 2t + 2. \]

Step 4: Isolate the Variable

Now, let's isolate \( t \) by subtracting \( 2t \) from both sides: \[ 3t - 2t - 4 = 2. \] Simplify to get: \[ t - 4 = 2. \]

Step 5: Solve for t

Finally, add 4 to both sides to solve for \( t \): \[ t = 6. \]

Step 6: Check the Solution for Validity

Substitute \( t = 6 \) back into the original expressions inside the logarithms to check if they are positive, ensuring the solution is valid: \( 3(6) - 4 = 14 \) and \( 6 + 1 = 7 \). Both are positive, so the solution is valid.

Key Concepts

Natural Logarithms

The natural logarithm, denoted by \( \ln \), is a special type of logarithm that uses the base \( e \), where \( e \approx 2.71828 \). It is commonly used in mathematics due to its natural occurrence in many mathematical contexts, especially involving growth processes.
Natural logarithms simplify many mathematical expressions and calculations. For example, solving equations involving exponentials often utilizes natural logarithms, because the inverse of an exponential function with base \( e \) is a natural logarithm.
The relationship between exponentials and natural logarithms can be expressed as:

• \( e^{\ln x} = x \)
• \( \ln(e^x) = x \)

This duality allows us to transform products into sums and divisions into differences, providing simpler forms which are easier to solve.

Properties of Logarithms

Logarithms have several properties that make them valuable tools for solving equations. For the given equation, one crucial property is \( \ln a - \ln b = \ln \left( \frac{a}{b} \right) \). This property allows us to combine or split logarithmic terms, simplifying complex expressions.
Let's look at some key properties:

• Product Property: \( \ln(ab) = \ln a + \ln b \)
• Quotient Property: \( \ln \left( \frac{a}{b} \right) = \ln a - \ln b \)
• Power Property: \( \ln (a^b) = b \ln a \)

These properties enable the transformation of equations, reducing their complexity and making them more manageable. In this exercise, the quotient property was used to simplify the expression \( \ln(3t-4) - \ln(t+1) \) into a single logarithmic expression of a rational function.

Solving Rational Equations

A rational equation is an equation containing fractions whose numerators and/or denominators include polynomials. The problem involves such an equation:
\[ \frac{3t-4}{t+1} = 2 \]Solving rational equations requires finding a common denominator or employing manipulation techniques such as cross multiplication.
Follow these steps to solve a rational equation:

• Step 1: Clear the fractions by multiplying every term by a common denominator, if possible.
• Step 2: Simplify the equation to a form without fractions.
• Step 3: Solve the resulting linear or quadratic equation.

In this instance, cross multiplication was directly applied to eliminate the fraction and simplify the manipulation.

Cross Multiplication

Cross multiplication is a method used to solve rational equations, particularly useful when the equation is in the form of a proportion, such as \( \frac{a}{b} = \frac{c}{d} \). In the given exercise, cross multiplication was used to solve:
\[ \frac{3t-4}{t+1} = 2 \]Here's how it works:

• Take both expressions on either side of the equation and multiply across the equals sign: \( (3t-4) \times 1 = 2 \times (t+1) \)
• This results in: \( 3t - 4 = 2t + 2 \)
• By isolating \( t \) and simplifying further, we solve for \( t \).

Cross multiplication is powerful because it removes denominators and leads directly to a linear equation, which is generally easier to solve.