Problem 40 - A Graphical Approach to Precalculus with Limits

Question

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$\log _{5}(8-3 x)=3$$

Step-by-Step Solution

Answer

The solution is \( x = -39 \).

Step 1: Understand the Equation

We are given the logarithmic equation \( \log_{5}(8 - 3x) = 3 \). Our task is to solve for \( x \). The equation states that the logarithm base 5 of \( 8 - 3x \) equals 3.

Step 2: Convert the Logarithmic Equation to Exponential Form

To solve the equation \( \log_{5}(8 - 3x) = 3 \), we convert it to its equivalent exponential form. The general rule is \( \log_b(a) = c \) can be rewritten as \( b^c = a \). Applying this, we get \( 5^3 = 8 - 3x \).

Step 3: Calculate the Exponent

Next, we calculate \( 5^3 \, \). The value is \( 125 \). So, our equation simplifies to \( 125 = 8 - 3x \).

Step 4: Solve for x

We need to isolate \( x \) in the equation \( 125 = 8 - 3x \). Start by subtracting 8 from both sides of the equation: \( 125 - 8 = -3x \). This simplifies to \( 117 = -3x \).

Step 5: Divide to Isolate x

Divide both sides of the equation by \(-3\) to solve for \( x \). This gives you \( x = \frac{117}{-3} \).

Step 6: Simplify the Solution

Simplify \( x = \frac{117}{-3} \) by performing the division: \( x = -39 \).

Step 7: Verify the Solution

Verify the solution by substituting \( x = -39 \) back into the original logarithmic equation to confirm the left-hand side equals the right-hand side after simplification.

Key Concepts

Exponential Form

When dealing with logarithmic equations, it's crucial to understand their relationship with exponentials. A logarithmic equation like \( \log_b(a) = c \) can be translated into an exponential form which is \( b^c = a \). This conversion is essential because it transforms the logarithmic equation into a form that is often easier to solve.

Let's break it down using an example: Consider the equation \( \log_{5}(8 - 3x) = 3 \). Here, the base \( b \) is 5, \( a \) is \( (8 - 3x) \), and \( c \) is 3. Converting to exponential form, it becomes \( 5^3 = 8 - 3x \).

This step is pivotal in simplifying the equation and isolates the term that can be easily manipulated. Understanding how to convert to exponential form can greatly aid in tackling various types of logarithmic problems.

Solving Equations

Once the logarithmic equation is in its exponential form, solving it becomes more straightforward. In our example, we have the equation \( 5^3 = 8 - 3x \). The next step involves computing the exponent.

• First, calculate \( 5^3 \). This means multiplying 5 by itself three times: \( 5 \times 5 = 25 \) and \( 25 \times 5 = 125 \).
• Substitute this value back into the equation, yielding \( 125 = 8 - 3x \).

To isolate \( x \), you perform algebraic operations:

• Subtract 8 from both sides: \( 125 - 8 = -3x \), simplifying to \( 117 = -3x \).
• Divide both sides by \(-3\) to solve for \( x \): \( x = \frac{117}{-3} \).
• Perform the division: \( x = -39 \).

By carefully following these steps, you can solve for the variable \( x \) precisely, ensuring an accurate solution.

Verification of Solutions

Verifying a solution is an important final step that ensures your solution is correct. It involves substituting the solution back into the original equation to check for consistency.

For the logarithmic equation \( \log_{5}(8 - 3x) = 3 \), we found \( x = -39 \). Plugging \( x = -39 \) back into the equation gives:

• Calculate \( 8 - 3(-39) \). First, find \( 3 \times -39 = -117 \). Then, compute \( 8 - (-117) = 8 + 117 = 125 \).

At this point, we need to verify if \( \log_{5}(125) = 3 \) holds true:

• Convert \( \log_{5}(125) \) to its exponential form: \( 5^3 = 125 \).
• Since \( 125 \) equals \( 5^3 \), the equation is verified and correct.

Verification helps ensure accuracy and confirms the solution satisfies the original equation, reinforcing confidence in your answer.