Logarithmic Equations

Definition

Logarithmic equations are equations where the variable appears within the argument of a logarithm. Solving these equations typically involves using the inverse relationship of exponents to “isolate” the variable.

Why It Matters

Solving these equations is essential for finding the scale or magnitude required in systems modeled by logarithms, such as determining the intensity of an earthquake, the pH of a solution, or the decibel level of a sound.

Core Concepts

• Guidelines for Solving Logarithmic Equations:
  1. Isolate the logarithmic term on one side of the equation. You may need to use the Laws of Logarithms to combine multiple terms.
  2. Write the equation in exponential form (or “exponentiate” both sides by raising the base to each side).
  3. Solve for the variable.
• One-to-One Property:
  • If $\log_a u = \log_a v$, then $u = v$.
    • How to read: “The statement that if the logarithm base a of u equals the logarithm base a of v, then u equals v.”
    • Meaning: Logarithm is one-to-one—equal logs with same base mean equal arguments (with u, v > 0).
• Extraneous Solutions: Logarithmic equations must be checked for validity; the argument of a logarithm ($M$ in $\log_a M$) must always be positive. Any solution that results in the log of a non-positive number is extraneous.

Connected Concepts

• Exponential Functions
• Logarithmic Functions
• Logarithmic Properties
• Exponential Growth Model
• Exponential Decay Model
• Inversion
• Feedback Loops
• Scale
• Map-Territory Relation