IB · Unit 1 · 1C Logarithms
Rewrite the number as a power of the base, then read the exponent. log with no base means base 10.
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What is a log
The answer to a log is an exponent!
log2 8 = 3 means 2³ = 8
Common log of a power of 10
1000 = 10³
log 1000 = 3 For powers of 10 you can count the zeros.
Other bases
8 = 2³, so log2 8 = 3
Log of 1
Any allowed base: logb 1 = 0 because b⁰ = 1.
Example: log2 1 = 0 because 2⁰ = 1.
Numbers less than 1
First write the decimal as a fraction.
0.5 = 12 = 2⁻¹
so log2 0.5 = −1
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.
Common logs of powers of 10. No calculator.
log10 1000
10 to what power is 1000? (count the zeros)
1000 has 3 zeros, so 10³ = 1000.
Answer: 3
log10 10 000
10 to what power is 10 000? (count the zeros)
10 000 has 4 zeros, so 10⁴ = 10 000.
Answer: 4
log10 1 000 000
10 to what power is 1 000 000? (count the zeros)
1 000 000 has 6 zeros, so 10⁶ = 1 000 000.
Answer: 6
log10 0.001
0.001 is said “one thousandth.”
one thousandth = 11000
11000 = 10⁻³
10 to what power is 0.001? −3
Answer: −3
log10 0.00001
0.00001 is said “one hundred-thousandth.”
one hundred-thousandth = 1100 000
1100 000 = 10⁻⁵
10 to what power is 0.00001? −5
Answer: −5
log10 10⁹
10⁹ in standard form is 1 000 000 000.
1 000 000 000 has 9 zeros.
10 to what power is 1 000 000 000? 9
Answer: 9
A base other than 10. Includes log of 0 and log of a negative number.
This means 2 to what power is 8?
2x = 8
log2 8 = 3
Answer: 3
This means 2 to what power is 16?
2x = 16
log2 16 = 4
Answer: 4
This means 3 to what power is 9?
3x = 9
log3 9 = 2
Answer: 2
This means 5 to what power is 25?
5x = 25
log5 25 = 2
Answer: 2
This means 4 to what power is 64?
4x = 64
log4 64 = 3
Answer: 3
This means 3 to what power is 1?
3x = 1
log3 1 = 0
Answer: 0
This means 5 to what power is 0?
5x = 0
Each time the exponent drops by 1, divide by 2.
| Power | Value |
|---|---|
| 2⁻⁵ | 132 |
| 2⁻⁴ | 116 |
| 2⁻³ | 18 |
| 2⁻² | 14 |
| 2⁻¹ | 12 |
| 2⁰ | 1 |
| 2¹ | 2 |
| 2² | 4 |
| 2³ | 8 |
| 2⁴ | 16 |
The values get closer and closer to 0, but they never equal 0.
The same is true for powers of 5.
No real number works. A power of 5 is never 0.
Answer: undefined
This means 2 to what power is −16?
2x = −16
Each time the exponent drops by 1, divide by 2.
| Power | Value |
|---|---|
| 2⁻⁵ | 132 |
| 2⁻⁴ | 116 |
| 2⁻³ | 18 |
| 2⁻² | 14 |
| 2⁻¹ | 12 |
| 2⁰ | 1 |
| 2¹ | 2 |
| 2² | 4 |
| 2³ | 8 |
| 2⁴ | 16 |
The values get closer and closer to 0, but they never equal 0.
The values are also never negative.
No real number works. A power of 2 is never −16.
Answer: undefined
The argument is a unit fraction less than 1.
This means 2 to what power is 14?
2x = 14
2⁻² = 14
log2 14 = −2
Answer: −2
This means 3 to what power is 127?
3x = 127
3⁻³ = 127
log3 127 = −3
Answer: −3
This means 2 to what power is 132?
2x = 132
2⁻⁵ = 132
log2 132 = −5
Answer: −5
This means 5 to what power is 1125?
5x = 1125
5⁻³ = 1125
log5 1125 = −3
Answer: −3
This means 4 to what power is 116?
4x = 116
4⁻² = 116
log4 116 = −2
Answer: −2
This means 8 to what power is 164?
8x = 164
8⁻² = 164
log8 164 = −2
Answer: −2
Decimals that are easy to change to fractions.
0.5 = 12
This means 2 to what power is 12?
2x = 12
2⁻¹ = 12
log2 0.5 = −1
Answer: −1
0.0625 = 116
This means 2 to what power is 116?
2x = 116
2⁻⁴ = 116
log2 0.0625 = −4
Answer: −4
0.04 = 125
This means 5 to what power is 125?
5x = 125
5⁻² = 125
log5 0.04 = −2
Answer: −2
0.25 = 14
This means 4 to what power is 14?
4x = 14
4⁻¹ = 14
log4 0.25 = −1
Answer: −1
0.125 = 18
This means 2 to what power is 18?
2x = 18
2⁻³ = 18
log2 0.125 = −3
Answer: −3
0.2 = 15
This means 5 to what power is 15?
5x = 15
5⁻¹ = 15
log5 0.2 = −1
Answer: −1
This section is not likely to be on an IB test. Rewrite the base and the argument as powers of the same prime.
This means 9 to what power is 13?
9x = 13
Rewrite 9 and 13 as powers of 3.
(3²)x = 3⁻¹
32x = 3⁻¹
2x = −1
x = −12
Answer: −12
This means 8 to what power is 12?
8x = 12
Rewrite 8 and 12 as powers of 2.
(2³)x = 2⁻¹
23x = 2⁻¹
3x = −1
x = −13
Answer: −13
This means \(\dfrac{3}{5}\) to what power is \(\dfrac{25}{9}\)?
\(\left(\dfrac{3}{5}\right)^{x} = \dfrac{25}{9}\)
\(\dfrac{25}{9} = \left(\dfrac{5}{3}\right)^{2}\)
\(\left(\dfrac{5}{3}\right)^{2} = \left(\dfrac{3}{5}\right)^{-2}\)
x = −2
Answer: −2
This means 27 to what power is 181?
27x = 181
Rewrite 27 and 181 as powers of 3.
(3³)x = 3⁻⁴
33x = 3⁻⁴
3x = −4
x = −43
Answer: −43
This means 8 to what power is 14?
8x = 14
Rewrite 8 and 14 as powers of 2.
(2³)x = 2⁻²
23x = 2⁻²
3x = −2
x = −23
Answer: −23
This means \(\dfrac{5}{2}\) to what power is \(\dfrac{8}{125}\)?
\(\left(\dfrac{5}{2}\right)^{x} = \dfrac{8}{125}\)
\(\dfrac{8}{125} = \left(\dfrac{2}{5}\right)^{3}\)
\(\left(\dfrac{2}{5}\right)^{3} = \left(\dfrac{5}{2}\right)^{-3}\)
x = −3
Answer: −3
Easy, Medium, and Hard. Hard 2 and Hard 3 are not in this section.
log10 100
10 to what power is 100? (count the zeros)
100 has 2 zeros, so 10² = 100.
Answer: 2
log10 0.01
0.01 is said “one hundredth.”
one hundredth = 1100
1100 = 10⁻²
10 to what power is 0.01? −2
Answer: −2
This means 5 to what power is 125?
5x = 125
log5 125 = 3
Answer: 3
This means 7 to what power is 0?
7x = 0
No real number works. A power of 7 is never 0.
Answer: undefined
This means 3 to what power is 19?
3x = 19
3⁻² = 19
log3 19 = −2
Answer: −2
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