Steply

Natural logarithms

Algebra 2 ยท 23 problems, easy to harder

โ–ถ Watch the first one solved

How we do it

  1. Take ln of both sides
  2. ln and e undo each other
  3. Remove the + 1
  4. Remove the 2
  5. Use a calculator
  6. Check

The steps for Solve e^(2x + 1) = 9. Every solution names its steps, so you always know what comes next.

Worked examples

Calculate ln(e^4) Easy

The question

Question: work out ln(eโด).

ln(e4)

e is a special number, like ฯ€.

It is about 2.718.

A log with base e is written ln.

We say "natural log".

e โ‰ˆ 2.718
ln = loge

1 What the log asks

A log is a power.

ln(eโด) asks: e to what power is eโด?

a log is a power

e to the power what is eโด?

e? = e4

2 Find the power

It is already e to a power: the power is 4.

ln and e undo each other.

ln and e undo each other

So the power is 4.

e4 = e4

So ln(eโด) = 4.

ln(e4) = 4

Let's look back.

First: what the log asks.

Last: find the power.

Answer
ln(e4) = 4

โ–ถ Watch it solved

Solve e^(2x) = 9 Medium

The question

Question: solve e to the power (2x) = 9.

x is up in the power.

e2x = 9

e is a special number, like ฯ€.

It is about 2.718.

A log with base e is written ln.

We say "natural log".

e โ‰ˆ 2.718
ln = loge

1 Take ln of both sides

To bring the power down, we use a log.

With e, we use ln.

We do it on both sides, so they stay equal.

ln(e2x) = ln 9

Careful!

x is up in the power.

Dividing by e does not bring it down.

Only a log does.

Careful: not x = 9 รท e โœ—

2 ln and e undo each other

ln asks: e to what power?

For e to the power k, the answer is k.

So ln and e undo each other.

ln(ek) = k

So the left side is just 2x.

2x = ln 9

3 Remove the 2

2x means 2 times x.

We want just x.

To remove times 2, we divide by 2.

We do it on both sides, so they stay equal.

2x2 = ln 92

2 divided by 2 is 1.

So just x is left.

So x = ln 9/2.

This is the exact answer.

x = ln 92

4 Use a calculator

With a calculator, ln 9 is about 2.1972.

Then 2.1972 รท 2 is about 1.099.

ln 9 โ‰ˆ 2.1972
2.1972 รท 2 โ‰ˆ 1.099

So x is about 1.099.

x โ‰ˆ 1.099

So x = ln 9/2, about 1.099 to three decimal places.

5 Check

Let's check with a calculator.

The power gives about 9.0.

That is 9.

It works!

Check: e2(1.099) โ‰ˆ 9.0

Let's look back.

First: take ln of both sides.

Then: ln and e undo each other.

Then: remove the 2.

Then: use a calculator.

Last: check.

Answer
x โ‰ˆ 1.099

โ–ถ Watch it solved

Solve e^(2x + 1) = 9 Harder

The question

Question: solve e to the power (2x + 1) = 9.

x is up in the power.

e2x + 1 = 9

e is a special number, like ฯ€.

It is about 2.718.

A log with base e is written ln.

We say "natural log".

e โ‰ˆ 2.718
ln = loge

1 Take ln of both sides

To bring the power down, we use a log.

With e, we use ln.

We do it on both sides, so they stay equal.

ln(e2x + 1) = ln 9

Careful!

x is up in the power.

Dividing by e does not bring it down.

Only a log does.

Careful: not x = 9 รท e โœ—

2 ln and e undo each other

ln asks: e to what power?

For e to the power k, the answer is k.

So ln and e undo each other.

ln(ek) = k

So the left side is just 2x + 1.

2x + 1 = ln 9

3 Remove the + 1

We want 2x by itself.

The + 1 is in the way.

To remove + 1, we take away 1.

We do it on both sides, so they stay equal.

2x + 1 โˆ’ 1 = ln 9 โˆ’ 1

+ 1 and โˆ’ 1 make zero.

Cross them out.

We keep it exact: 2x = ln 9 โˆ’ 1.

2x = ln 9 โˆ’ 1

4 Remove the 2

2x means 2 times x.

We want just x.

To remove times 2, we divide by 2.

We do it on both sides, so they stay equal.

2x2 = ln 9 โˆ’ 12

2 divided by 2 is 1.

So just x is left.

So x = (ln 9 โˆ’ 1)/2.

This is the exact answer.

x = ln 9 โˆ’ 12

5 Use a calculator

With a calculator, ln 9 is about 2.1972.

Then (2.1972 โˆ’ 1) รท 2 is about 0.599.

ln 9 โ‰ˆ 2.1972
(2.1972 โˆ’ 1) รท 2 โ‰ˆ 0.599

So x is about 0.599.

x โ‰ˆ 0.599

So x = (ln 9 โˆ’ 1)/2, about 0.599 to three decimal places.

6 Check

Let's check with a calculator.

The power gives about 9.0.

That is 9.

It works!

Check: e2(0.599) + 1 โ‰ˆ 9.0

Let's look back.

First: take ln of both sides.

Then: ln and e undo each other.

Then: remove the + 1.

Then: remove the 2.

Then: use a calculator.

Last: check.

Answer
x โ‰ˆ 0.599

โ–ถ Watch it solved

Practice

Try each one, then open the answer. Stuck? Open the problem to watch every step.

Easy

  1. ln(e^4)
    Answer
    ln(e4) = 4
  2. ln(e^2)
    Answer
    ln(e2) = 2
  3. ln(e^7)
    Answer
    ln(e7) = 7
  4. ln 1
    Answer
    ln(1) = 0
  5. e^x = 5
    Answer
    x โ‰ˆ 1.609
  6. e^x = 12
    Answer
    x โ‰ˆ 2.485
  7. e^x = 30
    Answer
    x โ‰ˆ 3.401
  8. ln(x) = 2
    Answer
    x โ‰ˆ 7.389
  9. ln(x) = 1
    Answer
    x โ‰ˆ 2.718

Medium

  1. e^(2x) = 9
    Answer
    x โ‰ˆ 1.099
  2. e^(3x) = 20
    Answer
    x โ‰ˆ 0.999
  3. e^(x + 1) = 10
    Answer
    x โ‰ˆ 1.303
  4. e^(x - 2) = 7
    Answer
    x โ‰ˆ 3.946
  5. ln(x - 1) = 3
    Answer
    x โ‰ˆ 21.086
  6. ln(x + 4) = 2
    Answer
    x โ‰ˆ 3.389
  7. ln(2x) = 1
    Answer
    x โ‰ˆ 1.359
  8. ln(3x) = 2
    Answer
    x โ‰ˆ 2.463

Harder

  1. e^(2x + 1) = 9
    Answer
    x โ‰ˆ 0.599
  2. e^(2x - 3) = 15
    Answer
    x โ‰ˆ 2.854
  3. ln(2x - 1) = 3
    Answer
    x โ‰ˆ 10.543
  4. ln(x) = -1
    Answer
    x โ‰ˆ 0.368
  5. e^(-x) = 4
    Answer
    x โ‰ˆ โˆ’1.386
  6. e^x = -3
    Answer
    No power of e is โˆ’3.