Problem 57 - A Graphical Approach to College Algebra

Question

Solve each logarithmic equation. Express all solutions in exact form. Support your solutions by using a calculator. $$\log _{7}(4 x)-\log _{7}(x+3)=\log _{7} x$$

Step-by-Step Solution

Answer

The solution is \( x = 1 \).

Step 1: Apply Logarithm Property

First, use the property of logarithm that states: \( \log_b(A) - \log_b(B) = \log_b\left(\frac{A}{B}\right) \). Therefore, rewrite the given equation as \( \log_{7}\left(\frac{4x}{x+3}\right) = \log_{7}x \).

Step 2: Equate Arguments

Since the logarithms are of the same base (7) and are equal, their arguments must also be equal: \( \frac{4x}{x+3} = x \).

Step 3: Solve the Equation

To eliminate the fraction, multiply both sides of the equation by \( x + 3 \), leading to:\[ 4x = x(x + 3) \].This expands to:\[ 4x = x^2 + 3x \].Rearrange it to form a quadratic equation:\[ x^2 - x = 0 \].

Step 4: Factor the Quadratic Equation

Factor the quadratic equation \( x^2 - x = 0 \) as \( x(x - 1) = 0 \). This gives the possible solutions: \( x = 0 \) or \( x = 1 \).

Step 5: Check Solutions in Original Equation

After finding potential solutions, check if they are valid within the original logarithmic equation. Substitute \( x = 0 \) which leads to undefined values (as you cannot take the logarithm of zero). Thus, \( x = 0 \) is not a valid solution.Substitute \( x = 1 \) into the original equation: \( \log_{7}(4 \times 1) - \log_{7}(1 + 3) = \log_{7}(1) \), simplifying to: \( \log_{7}(4) - \log_{7}(4) = 0 \), which is true.

Key Concepts

Properties of Logarithms

Logarithmic properties are fundamental tools that help simplify complex expressions into more manageable forms. A key property often used is the subtraction of logarithms with the same base, expressed as: \( \log_b(A) - \log_b(B) = \log_b\left(\frac{A}{B}\right) \). This property was crucial in reshaping the original equation \( \log _{7}(4 x)-\log _{7}(x+3)=\log _{7} x \) into a simpler form that could be solved more easily.
By using this property, we were able to condense the difference of two logarithms into a single logarithmic expression: \( \log_{7}\left(\frac{4x}{x+3}\right) = \log_{7}x \). This step moved us closer to a straightforward equation where we could equate the arguments.
In the world of logarithms, other useful properties include:

• The Product Property: \( \log_b(A \cdot B) = \log_b(A) + \log_b(B) \)
• The Power Property: \( \log_b(A^n) = n \cdot \log_b(A) \)

These properties provide the flexibility needed to manipulate and solve logarithmic expressions effectively.

Solving Quadratic Equations

Quadratic equations often appear in mathematical problems, including those involving logarithms, as shown in the solution process. Once the logarithmic expression \( \frac{4x}{x+3} = x \) was obtained, rearranging terms led to the quadratic equation: \( x^2 - x = 0 \).

This equation is easily solvable through factoring, yielding \( x(x - 1) = 0 \). Factoring involves finding two numbers that multiply to give the original quadratic while adding to zero here. This factorization resulted in potential solutions \( x = 0 \) or \( x = 1 \).
To factor a simple quadratic equation like \( ax^2 + bx + c = 0 \):

• Identify coefficients and the simplified equation form.
• Use factorization, the quadratic formula, or complete the square if necessary.

However, always verify solutions in the original context, as certain mathematical constraints may rule out some potential solutions, such as negative or zero values in logarithmic equations.

Logarithmic Functions

Logarithmic functions express a unique relationship between numbers, which involves an exponent and its resulting value. The general form \( y = \log_b(x) \) states that \( b^y = x \), where \( b \) is the base of the logarithm, and \( x \) is the number.
In solving logarithmic equations, understanding this inverse relationship helps transform complex expressions into solvable forms. For example, the given equation \( \log_{7}x \) implies that 7 must be raised to some power to result in \( x \).

When dealing with logarithmic functions, keep in mind:

• They are only defined for positive real numbers; i.e., \( x > 0 \).
• Knowing how to convert between logarithmic and exponential forms is helpful: \( y = \log_b(x) \rightarrow b^y = x \).

Logarithmic functions prove useful in a wide range of mathematical and real-world applications, including compound interest calculations, pH levels in chemistry, and even decibel levels in audio engineering. Understanding these functions broadens the scope of how we solve problems involving exponential and logarithmic relationships.