Solving Logarithmic Equations in College Algebra

Logarithmic Equations

Introduction to Logarithmic Equations

Logarithmic equations are equations that involve logarithms of algebraic expressions. Solving these equations is a key skill in College Algebra, requiring knowledge of logarithmic properties and algebraic manipulation. The following notes summarize the main concepts, properties, and solution strategies for logarithmic equations.

Key Properties of Logarithms

• Definition: The logarithm base b of x, written as $\log_b(x)$, is the exponent to which b must be raised to obtain x.

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

• Quotient Rule: $\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)$

• Power Rule: $\log_b(x^k) = k \log_b(x)$

• Change of Base Formula: $\log_b(x) = \frac{\log_a(x)}{\log_a(b)}$ for any positive base a ≠ 1.

Solving Logarithmic Equations

To solve logarithmic equations, use the properties of logarithms to combine or simplify terms, then solve for the variable. Always check that solutions are in the domain of the original logarithmic expressions (i.e., arguments must be positive).

• Step 1: Use logarithmic properties to combine or separate logarithms as needed.

• Step 2: If possible, rewrite the equation so that both sides have a single logarithm with the same base.

• Step 3: Set the arguments equal to each other (if bases and coefficients match), or exponentiate both sides to eliminate the logarithm.

• Step 4: Solve the resulting algebraic equation.

• Step 5: Check all solutions in the original equation to ensure they do not produce negative or zero arguments for any logarithm.

Examples from Worksheet

1. Equation: $2\log_2(x) + 3 = 0$

   • Step 1: Isolate the logarithm: $2\log_2(x) = -3$

   • Step 2: Divide by 2: $\log_2(x) = -\frac{3}{2}$

   • Step 3: Rewrite in exponential form: $x = 2^{-3/2}$

   • Example Solution: $x = \frac{1}{2^{3/2}} = \frac{1}{2\sqrt{2}}$

2. Equation: $\log_2(x) + \log_2(2x + 1) = 0$

   • Step 1: Use product rule: $\log_2(x(2x + 1)) = 0$

   • Step 2: Exponential form: $x(2x + 1) = 1$

   • Step 3: Solve quadratic: $2x^2 + x - 1 = 0$

   • Step 4: Factor: $(2x - 1)(x + 1) = 0$

   • Example Solution: $x = \frac{1}{2}$ or $x = -1$ (Check domain: $x > 0$ and $2x + 1 > 0$; only $x = \frac{1}{2}$ is valid)

3. Equation: $\log_2(x - 1) = 2\log_2(x - 1)$

   • Step 1: $\log_2(x - 1) = \log_2((x - 1)^2)$

   • Step 2: Set arguments equal: $x - 1 = (x - 1)^2$

   • Step 3: Solve: $x - 1 = x^2 - 2x + 1$

   • Step 4: $0 = x^2 - 3x + 2$

   • Step 5: Factor: $(x - 1)(x - 2) = 0$

   • Example Solution: $x = 1$ or $x = 2$ (Check domain: $x > 1$; only $x = 2$ is valid)

4. Equation: $2\ln(x + 3) = \ln(x + 1) + 3\ln2$

   • Step 1: Use power rule: $\ln((x + 3)^2) = \ln(x + 1) + \ln(2^3)$

   • Step 2: $\ln((x + 3)^2) = \ln(8(x + 1))$

   • Step 3: Set arguments equal: $(x + 3)^2 = 8(x + 1)$

   • Step 4: Expand: $x^2 + 6x + 9 = 8x + 8$

   • Step 5: $x^2-2x+6=0$

   • Step 6: Factor: $(x - 1)^2 = 0$

   • Example Solution: $x = 1$

5. Equation: $\log_2(2x - 9) = 2 - \log_2(x - 1)$

   • Step 1: Move all logs to one side: $\log_2(2x - 9) + \log_2(x - 1) = 2$

   • Step 2: Product rule: $\log_2((2x - 9)(x - 1)) = 2$

   • Step 3: Exponential form: $(2x - 9)(x - 1) = 2^2 = 4$

   • Step 4: Expand: $2x^2 - 2x - 9x + 9 = 4$

   • Step 5: $2x^2 - 11x + 13 = 0$

   • Step 6: Solve quadratic equation for $x$

6. Equation: $\log(x - 2) + \log(x - 3) = \log x^2$

   • Step 1: Product rule: $\log((x - 2)(x - 3)) = \log(x^2)$

   • Step 2: Set arguments equal: $(x - 2)(x - 3) = x^2$

   • Step 3: Expand: $x^2 - 5x + 6 = x^2$

   • Step 4: $-5x + 6 = 0$

   • Step 5: $x = \frac{6}{5}$

7. Equation: $\log_3\left(\frac{1}{x}\right) = 2$

   • Step 1: Rewrite: $\log_3(1) - \log_3(x) = 2$

   • Step 2: $0 - \log_3(x) = 2$

   • Step 3: $\log_3(x) = -2$

   • Step 4: Exponential form: $x = 3^{-2} = \frac{1}{9}$

Common Mistakes and Tips

• Always check the domain of the solution: the argument of any logarithm must be positive.

• When combining logarithms, ensure all terms have the same base.

• After solving, substitute solutions back into the original equation to verify validity.

Summary Table: Logarithmic Properties

Property	Formula	Example
Product Rule	$\log_b(xy) = \log_b(x) + \log_b(y)$	$\log_2(8) + \log_2(4) = \log_2(32)$
Quotient Rule	$\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)$	$\log_5(25) - \log_5(5) = \log_5(5)$
Power Rule	$\log_b(x^k) = k \log_b(x)$	$\log_3(9) = 2 \log_3(3)$
Change of Base	$\log_b(x) = \frac{\log_a(x)}{\log_a(b)}$	$\log_2(8) = \frac{\log_{10}(8)}{\log_{10}(2)}$