Problem 40 - A Graphical Approach to College Algebra

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 \( x = -39 \). Verification shows it is correct.

Step 1: Understand the Logarithmic Equation

We start with the given logarithmic equation: \( \log_{5}(8-3x) = 3 \). This tells us that the expression \(8-3x\) is the result of raising the base 5 to the power of 3.

Step 2: Rewrite the Equation Exponentially

Convert the logarithmic equation into an exponential form. The equation \( \log_{5}(8-3x) = 3 \) becomes \( 8-3x = 5^3 \).

Step 3: Calculate the Exponent

Evaluate the right side of the equation: \( 5^3 \). This gives us \( 125 \). Now the equation is \( 8 - 3x = 125 \).

Step 4: Solve for x

Isolate \( x \) by first subtracting 8 from both sides: \( 8 - 3x = 125 \) becomes \( -3x = 125 - 8 \), simplifying to \( -3x = 117 \). Then divide both sides by -3 to solve for \( x \): \( x = \frac{117}{-3} \), which simplifies to \( x = -39 \).

Step 5: Verify the Solution with a Calculator

Verify the solution by substituting \( x = -39 \) back into the original logarithmic equation. Calculate \( 8 - 3(-39) \), which simplifies to \( 8 + 117 = 125 \). Verify with a calculator: \( \log_{5}(125) \) should equal 3, as it matches \( 5^3 = 125 \). Calculation confirms the result is correct.

Key Concepts

Exponential Form

Understanding how to convert a logarithmic equation into exponential form is a crucial step in solving logarithmic equations. A logarithmic equation such as \( \log_{5}(8-3x) = 3 \) can be rewritten in exponential form. This transformation is based on the definition of a logarithm: if \( \log_{b}(a) = c \), it implies that \( b^c = a \). Thus, our equation becomes \( 8-3x = 5^3 \). By expressing the logarithmic function exponentially, we simplify the operation, moving from a complex logarithmic function to a straightforward arithmetic equation.

Base Conversion

The base in a logarithmic equation is the number that is raised to a power. In our example, this base is 5. Converting the base in logarithmic equations can often help simplify or rationalize the solution process. Though our problem does not require converting the base for simplification, it's important to know that such conversion could be useful if the problem was more complex. Base conversion involves using change of base formula

• If you have \( \log_{b}(x) \), it could be expressed in terms of natural or common logs: \( \frac{\log_{c}(x)}{\log_{c}(b)} \).

This technique proves extremely handy, especially when solving logarithmic equations with uncommon or inconvenient bases.

Solve for x

Once the equation is converted into its exponential form, solving for \( x \) becomes a matter of solving a linear equation. From our exponential form \( 8-3x = 125 \), isolate \( x \) by performing algebraic manipulations:

• First, subtract 8 from both sides to simplify: \( -3x = 125 - 8 \), which gives \( -3x = 117 \).
• Next, divide both sides by -3 to solve for \( x \): \( x = \frac{117}{-3} \).
• This calculation simplifies to \( x = -39 \).

These operations demonstrate the importance of understanding basic algebraic principles when solving for \( x \) in logarithmic equations.

Verify Solutions

Verification is an essential component in solving mathematical problems, ensuring the accuracy of your solution. To verify \( x = -39 \) in the context of our original logarithmic equation \( \log_{5}(8-3x) = 3 \), substitute the value back into the equation.

• Calculate \( 8 - 3(-39) \), which results in \( 8 + 117 = 125 \).
• Now, compute \( \log_{5}(125) \). According to the definition of a logarithm, since \( 5^3 = 125 \), \( \log_{5}(125) = 3 \).

Using a calculator confirms that substituting back into the equation holds true, validating our original solution. Verification not only confirms our work but also reinforces our confidence in the method used to solve the equation.