Problem 50 - College Algebra

Question

For the following exercises, solve each equation for \(x\). $$ \log _{3}(3 x)-\log _{3}(6)=\log _{3}(77) $$

Step-by-Step Solution

Answer

The solution is \( x = 154 \).

Step 1: Apply the Logarithm Difference Property

The equation given is \( \log_{3}(3x) - \log_{3}(6) = \log_{3}(77) \). Use the logarithmic property \( \log_{b}(m) - \log_{b}(n) = \log_{b}\left( \frac{m}{n} \right) \). This transforms the left side to \( \log_{3}\left( \frac{3x}{6} \right) \).

Step 2: Simplify the Expression

Simplify the expression \( \frac{3x}{6} \) to \( \frac{x}{2} \). The equation is now \( \log_{3}\left( \frac{x}{2} \right) = \log_{3}(77) \).

Step 3: Equate the Arguments of Logarithms

Since the bases of the logarithms are the same and the logs are equal, equate the arguments: \( \frac{x}{2} = 77 \).

Step 4: Solve for x

Multiply both sides of the equation \( \frac{x}{2} = 77 \) by 2 to solve for \( x \). This gives \( x = 154 \).

Key Concepts

Applying Logarithmic Properties

Understanding logarithmic properties is crucial when solving logarithmic equations. One such property is the difference property, which allows us to combine two logarithms that are subtracted. If you have \( \log_{b}(m) - \log_{b}(n) \), you can rewrite it as \( \log_{b}\left( \frac{m}{n} \right) \). This property simplifies the expression and makes equations easier to handle.

In the exercise, the original equation is \( \log_{3}(3x) - \log_{3}(6) \). By applying the difference property, this becomes \( \log_{3}\left( \frac{3x}{6} \right) \). This step is instrumental in reducing the equation to a simpler form, which then is simplified further in subsequent steps.

Equating Arguments of Logarithms

Once we have applied the logarithmic properties and simplified the equation, the next step is to equate the arguments. When you have an equation like \( \log_{b}(A) = \log_{b}(B) \), provided the bases are the same, you can deduce that the arguments must be equal: \( A = B \). This logical step is based on the fact that logarithmic functions are one-to-one.In our problem, after simplification, we are left with \( \log_{3}\left( \frac{x}{2} \right) = \log_{3}(77) \). Since the base on both sides of the equation is \( 3 \), it follows that \( \frac{x}{2} = 77 \). Equating the arguments reduces the problem to a more straightforward algebraic expression, allowing us to solve for \( x \) in the final step.

Solving Equations for Variables

The final step in solving the logarithmic equation is solving for the unknown variable. After equating the arguments, the problem often reduces to a simple algebraic equation, like \( \frac{x}{2} = 77 \). This step involves basic algebraic operations to isolate and find the value of the variable you are solving for.
To find \( x \), you multiply both sides of the equation by 2 to remove the fraction, yielding \( x = 154 \). This step reintroduces the solution into a practical context, completing the problem-solving process for the equation. Successfully solving for variables in logarithmic equations often requires comfort with basic algebra and arithmetic, thus reinforcing these skills in the learning process.