Problem 21 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 19-34: Solve the equation. $$ \log _{5}(x-2)=\log _{5}(3 x+7) $$

Step-by-Step Solution

Answer

There is no solution as both expressions must be positive for the logarithm.

Step 1: Equate the Expressions

Given the equation \( \log_{5}(x-2) = \log_{5}(3x+7) \), since the logarithms on both sides have the same base, we can equate the expressions inside the logarithms. This gives us the equation: \[ x-2 = 3x+7 \]

Step 2: Move Terms Involving x

To isolate \( x \), move all the terms involving \( x \) to one side of the equation. Subtract \( 3x \) from both sides: \[ x - 3x = 3x + 7 - 3x \] This simplifies to: \[ -2x = 7 + 2 \]

Step 3: Simplify the Equation

Now, simplify the equation obtained in the previous step: \[ -2x = 9 \]

Step 4: Solve for x

Divide both sides of the equation by \(-2\) to solve for \( x \): \[ x = \frac{9}{-2} \] This simplifies to: \[ x = -\frac{9}{2} \]

Step 5: Verify the Solution

To verify that \( x = -\frac{9}{2} \) is a valid solution, check if substituting \( x \) back into the original expressions inside the logarithms gives defined values. Substitute \( x \) into \( x - 2 \) and \( 3x + 7 \): \[ x - 2 = -\frac{9}{2} - 2 = -\frac{13}{2} \] \[ 3x + 7 = 3(-\frac{9}{2}) + 7 = -\frac{27}{2} + 7 = -\frac{13}{2} \] Since \( x - 2 \) and \( 3x + 7 \) must be greater than 0 and both resulted in a negative number, this solution is not valid.

Key Concepts

Solving Equations

Solving equations is a fundamental skill in algebra that involves finding the value of an unknown variable that makes the equation true. In the given problem with logarithms, we used a property that allows us to equate the expressions inside the log when the base is the same on both sides. This is because if \[ \log_b(A) = \log_b(B) \]then it follows that \( A = B \). Thus, our equation \[ \log_5(x-2) = \log_5(3x+7) \]transforms to \[ x - 2 = 3x + 7 \].

• Move all terms involving the unknown \( x \) to one side to isolate \( x \).
• Once all variables are on one side, simplify the terms.
• Divide by the coefficient of \( x \) to solve for \( x \).

In this method, each step contributes towards simplifying the equation while maintaining its equality, ultimately isolating the variable \( x \).

Logarithmic Properties

Logarithms have unique properties that make them useful for tackling specific types of equations. These properties simplify complex equations, especially when bases are equal, as seen in our problem. Some important properties of logarithms to remember are:

• Equality Property: If the logarithms on both sides have the same base, equate the terms within them.
• Product Property: \( \log_b(M \times N) = \log_b(M) + \log_b(N) \).
• Quotient Property: \( \log_b(\frac{M}{N}) = \log_b(M) - \log_b(N) \).
• Power Property: \( \log_b(M^p) = p \cdot \log_b(M) \).

These rules allow us to transform and simplify logarithmic expressions in a way that other algebraic tools cannot, making them especially powerful in equations involving exponential relationships.

Verification of Solutions

Verification is a crucial step in any equation-solving process. It ensures that the found solution actually satisfies the initial conditions of the problem.

• Involves substituting the solution back into the original equations.
• Check if the values inside the logarithms remain positive, as logarithms of non-positive numbers are undefined in real number settings.
• Ensures that any conclusion drawn is not due to an arithmetic mistake or the presence of extraneous solutions.

In our exercise, verification showed that \( x = -\frac{9}{2} \) leads to negative values inside the logarithms. Since logarithms are defined only for positive numbers, the solution was invalid. Thus, it’s essential to always verify the obtained solution to confirm its validity and mathematical feasibility.