Problem 49 - College Algebra

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 for \(x\) is \( \frac{3}{4} \).

Step 1: Apply Logarithm Property

Use the property of logarithms that states \( \ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right) \). Apply it to the left side of the equation: \[ \ln(3) - \ln(3 - 3x) = \ln \left( \frac{3}{3 - 3x} \right) \] So the equation becomes: \[ \ln \left( \frac{3}{3 - 3x} \right) = \ln(4) \]

Step 2: Remove the Logarithm

Since the natural logarithm on both sides of the equation is equal, we can set their arguments equal to each other: \[ \frac{3}{3 - 3x} = 4 \]

Step 3: Solve for \(x\)

To clear the fraction, perform cross-multiplication: \[ 3 = 4(3 - 3x) \]Simplify the right side: \[ 3 = 12 - 12x \]Add \(12x\) to both sides: \[ 12x + 3 = 12 \]Subtract 3 from both sides to isolate the term with \(x\): \[ 12x = 9 \]Finally, divide both sides by 12: \[ x = \frac{9}{12} = \frac{3}{4} \]

Key Concepts

Properties of Logarithms

Logarithmic properties are fundamental tools when working with logarithmic equations. They help simplify expressions and make solving equations much easier. One key property used in this exercise is the difference of logarithms, which states that \( \ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right) \). This property is extremely handy because it transforms the difference into a single logarithm.
By using this property in our problem, we transformed \( \ln(3) - \ln(3 - 3x) \) into \( \ln\left( \frac{3}{3 - 3x} \right) \). This simpler form is essential for easily setting both sides of the equation equal, allowing us to solve it.
Other useful properties of logarithms include:

• Product Property: \( \ln(a) + \ln(b) = \ln(ab) \)
• Power Property: \( \ln(a^b) = b \cdot \ln(a) \)

These properties enable us to manipulate and transform logarithmic expressions, revealing solutions hidden in their more complex forms.

Solving Equations

Solving logarithmic equations involves several steps, often starting with simplifying the equation using logarithmic properties. In the solution to this exercise, we applied the property of differences to achieve a single logarithmic expression. This simplification allowed us to directly compare arguments of the logarithms.
When both sides of a logarithmic equation contain the same logarithm base, like \( \ln \) in our example, you can equate the arguments directly. In our case, this meant rewriting the equation as \( \frac{3}{3-3x} = 4 \). This step removes the logarithms from the equation entirely, simplifying it to a more familiar algebraic form.
Next, solving the equation required some well-known algebraic techniques:

• Cross-multiplication to eliminate fractions.
• Isolating the variable \(x\) via algebraic manipulation (addition, subtraction, and division).

Breaking the problem into these steps helps tackle even complex logarithmic equations with clarity.

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a logarithm with a specific base \(e\), approximately equal to 2.718. It is one of the most commonly used logarithms in mathematics, particularly in calculus and higher-level mathematics.
The natural logarithm has unique properties, making it an integral part of solving logarithmic equations. The properties allow you to convert multiplicative relationships into additive ones, which is particularly useful in complex calculations. This simplification is evident in equations like \( \ln(a) = b \), which corresponds to the exponential form \( a = e^b \).
Understanding \( \ln \) is crucial because it frequently appears in real-world applications, such as calculating compound interest, analyzing exponential growth and decay, and solving differential equations. Its properties simplify many mathematical problems, turning tangled equations into simple, elegant solutions. By mastering these, students can unlock a more in-depth understanding of both pure and applied mathematical disciplines.