Lesson 5.16

Solving Exponential Equations (Different Bases)

When you can't rewrite both sides with the same base, take the log of both sides and use the power property to bring the unknown exponent down.

Introduction

— you can't make 50 a neat power of 3. The solution: take (or ) of both sides, then use the power property to bring down from the exponent.

Past Knowledge

Same-base solving (5.15), power property (5.13), log/ln evaluation (5.9).

Today's Goal

Solve exponential equations using log/ln and a calculator.

Future Success

5.18 applies this technique to real-world applications like half-life and doubling time.

Key Concepts

The Strategy

1

Isolate the exponential expression

2

Take or of both sides

3

Use power property to bring exponent down

4

Solve for the variable and use a calculator

Worked Examples

Example 1: Simple Case

Basic

Solve .

1

Take log of both sides

2

Power property

3

Solve

Example 2: With Base e

Intermediate

Solve .

1

Take (since base is )

2

Simplify ()

3

Divide

Example 3: Isolate First

Advanced

Solve .

1

Isolate the exponential

2

Recognize same base!

(Always check if same-base works after isolating!)

Common Pitfalls

Forgetting to Isolate

You must isolate the exponential term before taking the log. .

Rounding Too Early

Keep the exact form until the final step. Early rounding accumulates error.

Real-Life Applications

Most real-world exponential problems don't have neat same-base solutions. "How long until my $5,000 reaches $10,000 at 4%?" requires solving — and the answer is years.

Practice Quiz

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