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Logarithmic Equation

A logarithmic equation is an equation that contains at least one logarithm with a variable inside it. Solving one means finding the value of the variable that makes the equation true.

A logarithmic equation is an equation in which the unknown variable appears as part of the argument of a logarithmic function. To solve such equations, you typically use properties of logarithms to combine or simplify the logarithmic expressions, then convert the equation into exponential form. Because logarithms are only defined for positive arguments, any solution must be checked against the domain restrictions of the original equation.

Key Formula

log⁡b(x)=c⟺x=bc\log_b(x) = c \quad \Longleftrightarrow \quad x = b^c
Where:
  • bb = the base of the logarithm (must be positive and not equal to 1)
  • xx = the argument of the logarithm (must be positive)
  • cc = the value the logarithm equals

Worked Example

Problem: Solve the equation: log⁡2(x+3)+log⁡2(x)=4\log_2(x + 3) + \log_2(x) = 4
Step 1: Use the product rule of logarithms to combine the two logs into one.
log⁡2(x(x+3))=4\log_2(x(x + 3)) = 4
Step 2: Convert from logarithmic form to exponential form.
x(x+3)=24=16x(x + 3) = 2^4 = 16
Step 3: Expand and rearrange into a standard quadratic equation.
x2+3x−16=0x^2 + 3x - 16 = 0
Step 4: Solve the quadratic using the quadratic formula.
x=−3±9+642=−3±732x = \frac{-3 \pm \sqrt{9 + 64}}{2} = \frac{-3 \pm \sqrt{73}}{2}
Step 5: Check domain restrictions. Both log⁡2(x)\log_2(x) and log⁡2(x+3)\log_2(x+3) require positive arguments, so x>0x > 0. The solution x=−3−732x = \frac{-3 - \sqrt{73}}{2} is negative, so discard it.
x=−3+732≈2.77x = \frac{-3 + \sqrt{73}}{2} \approx 2.77
Answer: x=−3+732x = \dfrac{-3 + \sqrt{73}}{2}, which is approximately 2.772.77.

Why It Matters

Logarithmic equations show up whenever you need to solve for a quantity that appears inside an exponent—such as finding how long an investment takes to double at a given interest rate, or determining the magnitude of an earthquake on the Richter scale. They are also essential in chemistry (pH calculations) and acoustics (decibel levels).

Common Mistakes

Mistake: Forgetting to check that solutions produce positive arguments inside every logarithm.
Correction: Always substitute your answers back into the original equation. If any argument of a logarithm becomes zero or negative, that solution is extraneous and must be rejected.
Mistake: Dropping a logarithm when both sides already have one, without verifying the bases match.
Correction: You can only set log⁡b(A)=log⁡b(B)⇒A=B\log_b(A) = \log_b(B) \Rightarrow A = B when the bases are the same. If the bases differ, use the change of base formula first.

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