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Amy Ferguson Moncure

IB · Unit 1 · 1C Logarithms

Solve Log Equations

Ask: 10 to what power is the inside of the log? For ln, ask: e to what power? Then solve. Check that the argument of the log is positive.

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Key Ideas Math Facts Easy Medium Hard Mixed

The answer to a log is an exponent!

log2 8 = 3 means 2³ = 8

What to Remember

What is a log

The answer to a log is an exponent!

log2 8 = 3 means 2³ = 8

Undo a common log

log x = 4

This means 10 to what power is x? 10⁴ = x

x = 10 000

Undo a natural log

ln x = 3

This means e to what power is x? e³ = x

Log of an expression

log(5x−10) = 2

10² = 5x−10

x = 22

Check the argument

The input of a log must be positive. After you solve, check that the expression inside the log is greater than 0.

Math Facts

When a question writes log with no base, that means log10.

ln means loge, the natural logarithm.

A different base is written as a subscript, for example log2 8. This is read as log base 2 of 8.

Level 1

Easy

One log or ln equal to a number. Convert and evaluate.

Example
Solve for x.
log x = 4

log10 x = 4

This means 10 to what power is x?

10⁴ = x

x = 10 000

Answer: 10 000

1
Solve for x.
log x = 1
Show Answer

log10 x = 1

This means 10 to what power is x?

10¹ = x

x = 10

Answer: 10

2
Solve for x.
log x = 5
Show Answer

log10 x = 5

This means 10 to what power is x?

10⁵ = x

x = 100 000

Answer: 100 000

3
Solve for x.
ln x = 3
Show Answer

This means e to what power is x?

e³ = x

Answer: e³

4
Solve for x.
ln x = 0
Show Answer

This means e to what power is x?

e⁰ = x

x = 1

Answer: 1

5
Solve for x.
log x = −2
Show Answer

log10 x = −2

This means 10 to what power is x?

10⁻² = x

1100 = 0.01

x = 0.01

Answer: 0.01

Level 2

Medium

The log of a linear expression, or ln x equal to an expression in y.

Example
Solve for x.
log(5x−10) = 2

log10 (5x−10) = 2

This means 10 to what power is 5x−10?

10² = 5x−10

100 = 5x−10

110 = 5x

x = 22

Answer: 22

1
Solve for x.
log(4x+8) = 3
Show Answer

log10 (4x+8) = 3

This means 10 to what power is 4x+8?

10³ = 4x+8

1000 = 4x+8

992 = 4x

x = 248

Answer: 248

2
Solve for x.
log(x−6) = 1
Show Answer

log10 (x−6) = 1

This means 10 to what power is x−6?

10¹ = x−6

10 = x−6

x = 16

Answer: 16

3
Solve for x.
log(2x+6) = 0
Show Answer

log10 (2x+6) = 0

This means 10 to what power is 2x+6?

10⁰ = 2x+6

1 = 2x+6

2x = −5

x = −52

Check: 2(−52)+6 = 1 > 0. The log is defined.

Answer: −52

4
Find an expression for x.
ln x = y−2
Show Answer

This means e to what power is x?

ey−2 = x

Answer: ey−2

5
Find an expression for x.
ln x = 2y
Show Answer

This means e to what power is x?

e2y = x

Answer: e2y

Level 3

Hard

A negative log, ln of a linear expression, or a log equation that you rearrange first.

Example
Solve for x.
log(3x+1) = −1

log10 (3x+1) = −1

This means 10 to what power is 3x+1?

10⁻¹ = 3x+1

0.1 = 3x+1

3x = −0.9

x = −0.3

Check: 3(−0.3)+1 = 0.1 > 0. The log is defined.

Answer: −0.3

1
Solve for x.
log(4x−5) = −2
Show Answer

log10 (4x−5) = −2

This means 10 to what power is 4x−5?

10⁻² = 4x−5

0.01 = 4x−5

4x = 5.01

x = 1.2525

Check: 4(1.2525)−5 = 0.01 > 0. The log is defined.

Answer: 1.2525

2
Solve for x.
log(x+8) = −1
Show Answer

log10 (x+8) = −1

This means 10 to what power is x+8?

10⁻¹ = x+8

0.1 = x+8

x = −7.9

Check: −7.9+8 = 0.1 > 0. The log is defined.

Answer: −7.9

3
Find an expression for x.
ln(3x+6) = y+2
Show Answer

This means e to what power is 3x+6?

ey+2 = 3x+6

ey+2−6 = 3x

x = ey+2−63

Answer: ey+2−63

4
Find an expression for x.
ln(x−4) = 2y−1
Show Answer

This means e to what power is x−4?

e2y−1 = x−4

x = e2y−1+4

Answer: e2y−1+4

5
Solve for x.
1 + 3 log x = 7
Show Answer

3 log x = 6

log x = 2

log10 x = 2

This means 10 to what power is x?

10² = x

x = 100

Answer: 100

Mixed

Mixed Practice

Easy, Medium, and Hard.

1
Solve for x.
log x = 3
Show Answer

log10 x = 3

This means 10 to what power is x?

10³ = x

x = 1000

Answer: 1000

2
Solve for x.
ln x = 2
Show Answer

This means e to what power is x?

e² = x

Answer: e²

3
Solve for x.
log(x+15) = 2
Show Answer

log10 (x+15) = 2

This means 10 to what power is x+15?

10² = x+15

100 = x+15

x = 85

Answer: 85

4
Solve for x.
log x = −3
Show Answer

log10 x = −3

This means 10 to what power is x?

10⁻³ = x

11000 = 0.001

x = 0.001

Answer: 0.001

5
Solve for x.
log(3x−9) = 2
Show Answer

log10 (3x−9) = 2

This means 10 to what power is 3x−9?

10² = 3x−9

100 = 3x−9

109 = 3x

x = 1093

Answer: 1093

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