Solve Logarithmic Equations - Algebra-2

1. Fundamental Concepts

Definition: Logarithmic equations involve logarithms where the variable is in the argument of the logarithm function.
Properties: Key properties include the product rule ( $\log_b(xy) = \log_b(x) + \log_b(y)$ ), quotient rule ( $\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)$ ), and power rule ( $\log_b(x^p) = p\log_b(x)$ ).
One-to-One Property: If $\log_b(x) = \log_b(y)$ , then $x = y$ .

2. Key Concepts

Basic Rule: $$\log_b(x) + \log_b(y) = \log_b(xy)$$

Degree Preservation: The highest degree in the result matches input

Application: Used to solve complex exponential equations by converting them into simpler forms

3. Examples

1. Easy

Question: Solve $\log_2 (2x) = 5$

Solution:

Step 1: Use the logarithm-exponent inverse relationship. Convert $\log_2 (2x) = 5$ to exponential form: $2^5 = 2x$ .

Step 2: Calculate the exponential term: $32 = 2x$ , so $x = 16$ .

Step 3: Verify the domain: Substitute $x = 16$ into the argument $2x$ , we get $2 \times 16 = 32 > 0$ , which satisfies the domain requirement.

Conclusion: $x = 16$ is the solution.

2. Medium

Question: Solve $\log (x + 8) = -1$ (base 10, default for $\log$ )

Solution:

Step 1: Convert to exponential form. Since the base is 10, $\log (x + 8) = -1$ becomes $10^{-1} = x + 8$ .

Step 2: Calculate $10^{-1} = 0.1$ , so $0.1 = x + 8$ . Solve for $x$ : $x = 0.1 - 8 = -7.9$ .

Step 3: Verify the domain: Substitute $x = -7.9$ into the argument $x + 8$ , we get $-7.9 + 8 = 0.1 > 0$ , which is valid.

Conclusion: $x = -7.9$ (or $-\frac{79}{10}$ ) is the solution.

3. Difficult

Question: Solve $\log (7x + 1) - \log (x - 2) = 1$

Solution:

Step 1: Combine logarithms using the quotient rule. $\log (7x + 1) - \log (x - 2) = \log \left(\frac{7x + 1}{x - 2}\right) = 1$ .

Step 2: Convert to exponential form. Since $\log_{10} 10 = 1$ , the equation becomes $\frac{7x + 1}{x - 2} = 10^1 = 10$ .

Step 3: Solve the rational equation. Multiply both sides by $x - 2$ (note $x - 2 \neq 0$ , so $x \neq 2$ ): $7x + 1 = 10(x - 2)$ .

Expand the right-hand side: $7x + 1 = 10x - 20$ .

Rearrange terms: $3x = 21$ , so $x = 7$ .

Step 4: Verify the domain and original equation:

Check the argument of the original logarithms: $7x + 1 = 7 \times 7 + 1 = 50 > 0$ ; $x - 2 = 7 - 2 = 5 > 0$ (satisfies domain).

Substitute into the original equation: Left-hand side = $\log 50 - \log 5 = \log \left(\frac{50}{5}\right) = \log 10 = 1$ , which equals the right-hand side.

Conclusion: $x = 7$ is the solution.

4. Problem-Solving Techniques

Visual Strategy: Use a flowchart to outline the steps involved in solving logarithmic equations.
Error-Proofing: Always check the domain of the logarithmic functions before and after solving to ensure solutions are valid.
Concept Reinforcement: Practice with a variety of problems that involve different properties of logarithms.