Logarithm as the Inverse Function

Lesson 1 of 2: Definition and Conversion

In this lesson:

• Define logarithm as the inverse of exponentiation
• Evaluate logarithms using the "what exponent?" question
• Convert between logarithmic and exponential forms

What You Will Learn Today

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

1. State that is the inverse of
2. Convert between exponential form and logarithmic form
3. Evaluate logarithmic expressions by converting to the "what exponent?" question

Finding the Inverse of Exponentiation

Every operation has an inverse that undoes it:

• Squaring → square root:
• Adding 5 → subtracting 5:
• Multiplying by → dividing by :

Raising to a power: ???

The Logarithm as a Question

asks: "2 to what power gives 8?"

Since , the answer is — so .

Key idea: always asks " to what power gives ?" The logarithm IS the exponent that answers that question.

Exponential and Logarithm: Mirror Images

and are reflections over

Evaluating Logarithms: The What-Exponent Method

Evaluate :

"2 to what power gives 8?" → →

Step by step:

1. Identify the base:
2. Ask: "2 to what power gives ?"
3. Answer from memory: , so the log equals

More Evaluations Across Different Bases

Evaluate : "3 to what power gives 81?" → → answer: 4

Evaluate : "10 to what power gives 1000?" → → answer: 3

Same question, any base — the method never changes

Formal Definition: Two Equivalent Forms

These two statements say exactly the same thing:

Quick Check: Evaluate These Logarithms

Evaluate without a calculator:

Ask the "what exponent?" question for each before the next slide

Same Relation, Two Different Notations

Answers: because · because · because

Going forward: every logarithmic statement has a twin in exponential form — and vice versa.

Converting Logarithmic and Exponential Form

The base stays the base. The log value becomes the exponent. The argument becomes the result.

Converting Both Directions: Four Examples

Two Universal Facts About Any Logarithm

For any valid base :

These hold for every base — no exceptions

Negative Logs: When Arguments Are Fractions

Evaluate :

"2 to what power gives ?"

The logarithm CAN be negative — only the argument must be positive

Quick Check: Convert These Statements

Convert to exponential form:

Convert to logarithmic form:

Write all three before the next slide

Guided Practice: Convert in Both Directions

Convert to logarithmic form:

The base is , the exponent is , the result is — so:

Your turn — convert to logarithmic form:

Write your answer before the next slide reveals it

Your Turn: Mixed Conversion Problems

Convert as indicated:

1. → exponential form
2. → evaluate
3. → logarithmic form
4. → evaluate
5. → logarithmic form
6. → solve for

Pause and complete all six

Mixed Conversion Problems: Answers Revealed

2. because
4. (universal property)

Problems 2, 4, and 5 involve negative or "extended" logarithms — all valid

Key Takeaways: Logarithms as Inverses

✓ is the inverse of
✓
✓ and

Argument ; log can be negative

Up Next: Lesson Two Preview

You can now evaluate logarithms and convert between forms.

In Lesson 2:

• Composition properties: and
• Solving exponential equations:
• Solving logarithmic equations: