Solving Exponential Equations with Logarithms

Lesson 1 of 2: The Three-Step Procedure

In this lesson:

• Isolate the exponential term
• Convert to logarithmic form
• Solve for the variable using technology

What You Will Learn Today

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

1. Isolate the exponential term by dividing both sides by
2. Rewrite as
3. Solve for by dividing both sides by
4. Evaluate logarithmic expressions using a calculator

When Does the Population Hit One Million?

You know this model:

• Guess and check: try ? ? ?
• Each guess requires a calculator and still might miss

There must be a faster way — and there is.

Overview: The Three-Step Solution Method

Every equation of the form follows the same path:

1. Isolate — divide both sides by
2. Log — rewrite in logarithmic form
3. Divide — divide both sides by

Same structure, every time — regardless of the base

Step 1: Isolate the Exponential Term

Why this first? The logarithm only applies to the exponential term — not the whole left side.

This is just like solving : divide by 5 first

Applying Isolation to a Worked Example

Solve:

Step 1 only — divide both sides by 500:

Now the exponential term is alone. We're ready for Step 2.

What Goes Wrong Without Isolating First

Skipping Step 1: Apply log to directly

This requires the product rule — extra complexity we don't need.

Always isolate first. Then apply the logarithm.

Your Turn: Complete the Isolation Step

Your turn — Step 1 only:

What does the equation look like after dividing both sides by 300?

Write it down before the next slide

The Logarithm as an Inverse Operation

If , then

Step 2: Convert to Logarithmic Form

From Step 1 result:

Apply the definition: The exponent is what we want, so:

The exponent becomes the output of the logarithm

Base 10 Example: Logarithmic Form

From:

Apply log base 10 to both sides:

Written with standard notation:

On your calculator: the log key means base 10

Base Example: Logarithmic Form

From:

Apply the natural logarithm to both sides:

On your calculator: the ln key means log base

— the natural base for continuous growth

Calculator Keys for Each Base

Keep this in mind — it tells you exactly which key to press

Base 2 Example: Logarithmic Form and Clean Answer

From:

Since , we know :

When the logarithm is a whole number, no calculator needed

Your Turn: Write Logarithmic Form

Write in logarithmic form:

For problem 2: can you find the exact value mentally?

Transition: From Log Form to Solving for

So far in Chunk 1:

• ✓ Isolate the exponential (divide by )
• ✓ Convert to log form:

Next: We have — but we want .

One more algebraic step, then the calculator finishes the job

Step 3: Divide Both Sides by

After Step 2:

Divide both sides by :

The log gives you , not — always divide by

Worked Example: Full Solution with Base 10

Solve:

Step 1:

Step 2:

Step 3:

Calculator: , so

Calculator Keystrokes for All Three Bases

• Base 10: press log, enter the number →
• Base : press ln, enter the number →
• Base 2: use change-of-base → or

Worked Example: Full Solution with Base

Solve:

Step 1:

Step 2:

Step 3:

Calculator: , so

Full Example: Base 2 with Change-of-Base

Solve:

Step 1:

Step 2:

Step 3:

Decimal Answers Are Normal and Correct

After solving, you'll almost always get a decimal:

• years (not exactly 6)
• years (not exactly 22)
• hours (not exactly 17)

Round to context — report " years" or "between 21 and 22 years"

Guided Practice: All Three Steps

Solve:

Step 1 is done for you:

Step 2: Write in logarithmic form — what does equal?

Step 3: Solve for using your calculator

Complete Steps 2 and 3, then check the next slide

Independent Practice: One Per Base

Solve each equation completely. Show all three steps.

Pause and work through all three before the next slide

Practice Answers: One Per Base

Problem 3: is a recognizable power — clean answer

Three Steps and Four Warnings to Remember

✓ Isolate first — divide by
✓ Log form:
✓ Divide by — the log gives , not

Isolate before taking log
log = base 10, ln = base
Decimal answers are correct
Compute first, then take log

Coming Up: Applying the Procedure to Real Contexts

In Lesson 2, apply the three steps to:

• Doubling time — when does a quantity double?
• Half-life — when does decay reach a target?
• Mixed models — growth, investment, decay

Same three steps — interpretation is the new skill