Solving Logarithmic Equations in College Algebra

Exponential and Logarithmic Functions

Solving Logarithmic Equations

Logarithmic equations are equations that involve logarithms of algebraic expressions. Solving these equations often requires using properties of logarithms to combine terms and isolate the variable.

• Key Point 1: Properties of Logarithms are essential for simplifying and solving equations. The most common properties are:

  • Product Rule: $\log_b(M) + \log_b(N) = \log_b(MN)$

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

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

• Key Point 2: Isolating the Variable often involves combining logarithmic terms and then rewriting the equation in exponential form to solve for the unknown.

• Key Point 3: Checking Solutions is important because logarithms are only defined for positive arguments. Any solution that makes the argument of a logarithm zero or negative must be excluded.

Example: Solving a Logarithmic Equation

Consider the equation:

$\log_2(x - 1) + \log_2(x - 4) = 1$

1. Combine the logarithms using the product rule: $\log_2((x - 1)(x - 4)) = 1$

2. Rewrite the equation in exponential form: $(x - 1)(x - 4) = 2^1$ $(x - 1)(x - 4) = 2$

3. Expand and solve the quadratic equation: $x^2 - 5x + 4 = 2$ $x^2 - 5x + 2 = 0$

4. Solve for $x$ using the quadratic formula: $x = \frac{5 \pm \sqrt{(-5)^2 - 4 \cdot 1 \cdot 2}}{2}$ $x = \frac{5 \pm \sqrt{25 - 8}}{2}$ $x = \frac{5 \pm \sqrt{17}}{2}$

5. Check for extraneous solutions: Substitute each value back into the original equation to ensure the arguments of the logarithms are positive.

Additional info: The original question asks for the solution in terms of common or natural logarithms if necessary. In this case, the solution is exact and does not require conversion to another logarithm base.