Problem 339 - Precalculus

Question

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 Logarithmic Properties

Apply the property \( ln(a) - ln(b) = ln\left(\frac{a}{b}\right) \) to rewrite the equation: \( ln\left(\frac{3}{3-3x}\right) = ln(4) \).

Step 2: Exponentiate to Eliminate Logs

To eliminate the logarithms, exponentiate both sides using base \(e\): If \( ln(a) = ln(b) \), then \( a = b \). Therefore, \( \frac{3}{3-3x} = 4 \).

Step 3: Clear the Fraction

Multiply both sides by \(3-3x\) to clear the fraction: \(3 = 4(3-3x) \). This becomes \( 3 = 12 - 12x \).

Step 4: Solve for x

Isolate \(x\) by first subtracting 12 from both sides: \(3 - 12 = -12x\). Simplify to get \(-9 = -12x\). Divide both sides by -12 to solve for \( x \): \( x = \frac{-9}{-12} \). Simplify the fraction to get \( x = \frac{3}{4} \).

Key Concepts

Logarithmic Properties

When solving logarithmic equations, understanding the properties of logarithms is crucial. One key property used frequently is the difference of two logarithms. This property states:

• For any positive numbers, \( ext{ln}(a) - ext{ln}(b) = ext{ln}\left( \frac{a}{b} \right) \).

This property allows you to combine two logarithms into one, which can simplify the equation significantly.
By transforming the original equation, we convert a difference into a single logarithmic term, making it easier to manipulate. Recognizing these properties can often provide the quickest path to simplify and solve the given equation.

Exponentiation

Exponentiation is a valuable tool when dealing with logs because it helps remove them and transition into simpler algebraic forms. Through this process, the logarithmic equation can be solved using basic algebraic methods.
When two expressions are equal with the natural logarithm, such as \( ext{ln}(a) = ext{ln}(b) \), we can use the property of exponentiation base \(e\) which results in \(a = b\).

• This allows us to eliminate the logarithms.
• We are left with an expression that is generally easier to handle.

Exponentiating both sides transitions the complexity inside the logarithms to a straightforward algebraic equation.

Fraction Simplification

Simplifying fractions is an essential skill in algebra, especially in equations involving rational expressions. Often, by clearing fractions, we can reduce complex equations to simpler forms. In the context of this problem:

• First, you need to multiply both sides of the fraction by the denominator.
• This action eliminates the fraction form, turning it into a linear equation.

So, when you have an equation like \( \frac{3}{3-3x} = 4 \), clearing the fraction by multiplying both sides by \(3 - 3x\) produces: \(3 = 4(3 - 3x)\).
Through this step, the problem changes into solving a simple linear equation.

Equation Solving Steps

Solving a logarithmic equation involves various methods, which can be seen as a sequence of logical steps.

• Firstly, utilize logarithmic properties to condense or simplify the equations.
• Next, if possible, exponentiate to remove logarithms altogether.
• Then, address any fractions by multiplying through by the denominator to clear them.

Finally, isolate the variable by performing algebraic operations like addition, subtraction, or division on both sides of the equation.
In our example:

• Subtract 12 from both sides to simplify: \(3 - 12 = -12x\).
• Then, divide by \(-12\) to find \(x\).

By following these steps methodically, you can solve the equation and find that \(x = \frac{3}{4}\). This structured approach ensures you can handle more complex logarithmic equations with confidence.