Problem 49 - College Algebra with Corequisite Support

Question

For the following exercises, solve each equation for \(x\). \(\ln (3)-\ln (3-3 x)=\ln (4)\)

Step-by-Step Solution

Answer

The solution is \(x = \frac{3}{4}\).

Step 1: Use the Properties of Logarithms

We start by using the property of logarithms that says \( (a) - (b) = \left( \frac{a}{b} \right)\). So, the equation becomes \left( \frac{3}{3-3x} \right) = (4).

Step 2: Eliminate the Logarithm

To eliminate the logarithm, we can set the inside of the logs equal to each other: \( \frac{3}{3-3x} = 4 \).

Step 3: Solve the Equation for x

First, cross-multiply to eliminate the fraction: \( 3 = 4(3-3x) \). Simplify to \( 3 = 12 - 12x \).

Step 4: Isolate x

Move the constant term to the other side: \( 3 - 12 = -12x \), which simplifies to \( -9 = -12x \).

Step 5: Solve for x

Divide both sides by -12 to solve for \(x\): \(x = \frac{-9}{-12} \). Simplify this to \(x = \frac{3}{4} \).

Key Concepts

Logarithmic Equations

Logarithmic equations are mathematical expressions that involve logarithms with variables. These equations can look complex at first, but they follow straightforward rules. A typical logarithmic equation involves expressions like \( \log_a(x) \) or \( \ln(x) \), where \( \log \) refers to the logarithm and \( a \) is the base. The key to solving logarithmic equations is to work with the properties of logarithms, which can help simplify the equation. These properties include product, quotient, and power rules. Solving logarithmic equations usually involves rewriting them in exponential form to eliminate the logarithms and then generating simpler equations that can be solved using basic algebraic techniques.

Properties of Logarithms

The properties of logarithms make it possible to solve complex equations by simplifying them. These properties include:

• Product Rule: \( \log_b(MN) = \log_bM + \log_bN \). This helps combine two logarithms into one.
• Quotient Rule: \( \log_b\left(\frac{M}{N}\right) = \log_bM - \log_bN \). This rule is particularly useful when you have a difference of two logs, as seen in our problem.
• Power Rule: \( \log_b(M^k) = k\log_bM \). This helps in dealing with exponents within a log.

By applying these properties, you can convert differences or sums within logs into more manageable expressions. In our exercise, the Quotient Rule was used to transform the difference of logs into a single log, easing further calculations.

Cross Multiplication

Cross multiplication is a technique to solve equations that involve fractions. It allows you to remove fractions by multiplying across the equal sign. If you have an equation of the form \( \frac{a}{b} = \frac{c}{d} \), cross multiplication means you compute \( ad = bc \).

In the given problem, after applying logarithmic properties, our equation turned into \( \frac{3}{3-3x} = 4 \). By applying cross multiplication, the equation becomes \( 3 = 4(3-3x) \). This step helps in getting rid of the fraction, making the equation straightforward to solve using simple algebraic manipulation.

Algebraic Manipulation

Algebraic manipulation involves using algebraic techniques to simplify equations and solve for variables. After setting up an equation by eliminating logs or using cross multiplication, algebraic manipulation is key to finding the solution.

In the given exercise, this manipulation involves expanding the terms and isolating the variable \(x\). Starting with the equation \( 3 = 4(3-3x) \), you expand it to get \( 3 = 12 - 12x \).

Then, by moving the terms and simplifying, you arrive at \( -9 = -12x \). Finally, divide both sides by \(-12\) to solve for \(x\).

• Moving Terms: helps in getting constants on one side and terms with the variable on the other.
• Division: allows isolation of the variable by dividing both sides by the coefficient of the variable.

These steps demonstrate how algebraic manipulation turns a complex problem into a manageable one, leading to the final solution, \( x = \frac{3}{4} \).