Solving Equations with Logarithms

Learn the critical skill of identifying which type of equation you're dealing with before solving. Understand the three cases: exponential equations with same base, exponential equations requiring logarithms, and logarithmic equations. Master the decision flowcharts that determine your solution strategy.

Lesson 3.8 of Exponents & Logarithms in Math 30-1: Alberta curriculum lessons.

What you'll learn in this lesson

• The Decision Framework, Three Cases. Learn the critical skill of identifying which type of equation you're dealing with before solving. Understand the three cases: exponential equations with same base, exponential equations requiring logarithms, and logarithmic equations. Master the decision flowcharts that determine your solution strategy.
• Case 1 & Case 2: Exponential Equations. Work through detailed examples of exponential equations using both methods. Case 1: Convert both sides to the same base and equate exponents. Case 2: Use logarithms when bases can't be matched. Each example includes step-by-step solutions and verification checks.
• Case 3: Logarithmic Equations. Master logarithmic equations where the variable appears inside the argument. State restrictions, simplify using log laws, compare arguments or convert to exponential form, and always check solutions against restrictions to identify and reject extraneous roots.
• Advanced Logarithmic Equations. Tackle advanced logarithmic equations requiring algebraic manipulation. Learn to handle quadratic-style log equations using substitution, solve nested logarithms by working from outside in, and express variables in terms of other variables using log laws.

Inside the lesson: a free preview

The Big Picture

There are two types of equations:

Exponential Equation. Variable in the exponent: 5^x = 150

Logarithmic Equation. Variable inside the log: (x) - (20) = (0.2)

Each type has different solving methods — that's where the three cases come in. Cases 1 and 2 are for exponential equations. Case 3 is for logarithmic equations.

Your job. Identify which case before you start solving.

Exponential Equations — The Flowchart

How to solve. Step 1: Isolate the exponential.

Step 2: Can both sides be written with the same base? YES → Case 1. NO → Case 2.

Case 1 — Same Base

When to use. Both sides are powers of the same number (2, 3, 5, 10, etc.)

Steps. Change both sides to same base → Set exponents equal → Solve

Example. 9^(x-2) = (1)/(243) becomes 3^(2(x-2)) = 3^(-5), so 2(x-2) = -5

Case 2 — Can't Find Same Base

When to use. One side isn't a clean power (like 5^x = 150)

Method A — Direct conversion. c^x = N becomes x = _c(N)

Method B — Take log of both sides. Apply log → Use Power Law → Solve. Better when variables are on both sides.

Worked example: a: Solve 9^(x-2) = (1)/(243)

Step 1: Identify the common base. Both 9 and 243 are powers of 3.

9 = 3^2, so 9^(x-2) = (3^2)^(x-2) = 3^(2(x-2)) = 3^(2x-4)

243 = 3^5, so (1)/(243) = 3^(-5)

Step 2: Rewrite the equation. 3^(2x-4) = 3^(-5)

Step 3: Set exponents equal. 2x - 4 = -5

2x = -1

x = -(1)/(2)

✓ Verify. 9^(-(1)/(2)-2) = 9^(-(5)/(2)) = (3^2)^(-(5)/(2)) = 3^(-5) = (1)/(243) ✓

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