Solving with Logarithms

Lesson 2 of 2: Properties and Equations

In this lesson:

• Use composition properties to simplify expressions
• Solve exponential equations by applying logarithms
• Solve logarithmic equations by applying exponentiation

What You Will Learn Today

By the end of this lesson, you should be able to:

4. Use the composition properties: and
5. Solve exponential equations:
6. Solve logarithmic equations:

When Inverse Functions Cancel Each Other

You know from inverse function theory: if and are inverses, then:

Applied here: and — so each undoes the other.

Inverse Functions Cancel in Both Directions

Order 1: Apply exponential, then logarithm:

Order 2: Apply logarithm, then exponential:

Both hold because and are inverse functions

First Property: Exponential Undoes the Logarithm

Examples:

The exponent and the logarithm are the same base — they cancel

Applying the First Cancellation Property

Simplify :

The exponent is — same base as , so they cancel:

Simplify : same reasoning → result is

Check: does the base of the exponent match the base of the log?

Second Property: Logarithm Undoes the Exponential

Examples:

The logarithm returns the exponent — even when the exponent is negative

Both Cancellation Properties Side by Side

Same base → they cancel and return the inner value

Quick Check: Apply Cancellation Properties

Simplify:

For each: identify the bases — do they match? What does that tell you?

Using Inverse Properties to Solve Equations

Answers: · ·

• If — unknown is an exponent → apply log to both sides
• If — unknown is an argument → apply exponentiation to both sides

Solving Exponential Equations with Logarithms

When the unknown is in the exponent — apply to both sides:

The logarithm "brings down" the exponent, making the subject

Worked Example: Solve Two to the X

Solve:

By inspection: , so

Verify using logarithms:

Since , the what-exponent question confirms ✓

Clean answers like this confirm the logarithm method works

Worked Example: Solve Three to the X

Solve:

No integer power of 3 gives exactly 20 — can't solve by inspection.

Apply log:

Change-of-base:

Solving Logarithmic Equations with Exponentiation

• Unknown in exponent: apply →
• Unknown in argument: apply →
• Identify the structure before choosing the operation

Worked Example: Solve a Log Equation

Solve:

The unknown is inside the logarithm — apply to both sides:

Exponentiation cancels the logarithm; steps out

Worked Example: Log with Expression Inside

Solve:

The unknown is inside the log — exponentiate both sides base 2:

First exponentiate to escape the log, then solve the remaining linear equation

Guided Practice: Exponential Equation Challenge

Solve:

Step 1: Apply to both sides:

Step 2: Change-of-base — choose or :

Complete Step 2 with a calculator before the next slide

Independent Practice: Mixed Equation Types

Solve each equation completely:

Label your move for each: "apply log" or "apply exponentiation"

Practice Answers: Check Your Work

2. because (clean answer)

Key Takeaways: Using Inverse Relationships

✓ and
✓ → apply →
✓ → apply →

Bases must match for cancellation
Argument — check after solving

Up Next: Logarithm Graphs and Properties

You can now solve both exponential and logarithmic equations.

Coming up:

• Graph and analyze its properties
• Product, quotient, and power rules for logarithms
• Connecting exponential and logarithmic models in context

The inverse relationship is the thread connecting all of logarithm theory