Logarithms and lg
lg x is the exponent that 10 must be raised to in order to give x: lg 1000 = 3, because 10³ = 1000. So 10ˣ = 500 has the solution x = lg 500 ≈ 2.70. With another base, we take lg of both sides.
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What does lg mean?
lg 100 asks: 10 to the power of what is 100?
lg 100 = 2 because 10² = 100
| Power of 10 | Value | Logarithm |
|---|---|---|
| 10⁻¹ | 0.1 | lg 0.1 = −1 |
| 10⁰ | 1 | lg 1 = 0 |
| 10¹ | 10 | lg 10 = 1 |
| 10² | 100 | lg 100 = 2 |
| 10³ | 1000 | lg 1000 = 3 |
500 lies between 100 and 1000, so lg 500 ≈ 2.699 lies between 2 and 3. On a calculator the button is called log.
The logarithm laws
Logarithms are exponents, so they follow the laws of exponents. 10² · 10³ = 10⁵, so lg 100 + lg 1000 = 2 + 3 = 5.
| Law | In words | Example |
|---|---|---|
| lg(a · b) = lg a + lg b | a product becomes a sum | lg(100 · 1000) = 2 + 3 = 5 |
| lg(a / b) = lg a − lg b | a quotient becomes a difference | lg(1000 / 10) = 3 − 1 = 2 |
| lg(aᵖ) = p · lg a | the exponent comes down as a factor | lg(2³) = 3 · lg 2 ≈ 0.903 |
The laws hold for positive numbers a and b. The third law brings x down from the exponent. Then we can divide by whatever is next to x.
Solving equations with lg
We want to get x on its own. x is in the exponent.
- Get the power on its own on one side.
- Take lg of both sides. Equal numbers have equal logarithms.
- Use lg(aˣ) = x · lg a to bring x down.
- Divide both sides by the logarithm next to x.
Example
Example 1: 10ˣ = 500
We want to get x on its own. The base is 10, so lg gives the answer directly.
What exponent must 10 have to give 500? That is lg 500.
x = lg 500
The calculator gives an answer between 2 and 3.
x ≈ 2.699
Rounded to two decimal places:
x ≈ 2.70
10 to the power of 2.699 is about 500. That is correct.
Example 2: 3 · 2ˣ = 48
We want to get x on its own. First 2ˣ must be on its own, then we bring x down with lg.
We divide both sides by 3. 48 ÷ 3 = 16.
2ˣ = 16
We take lg of both sides.
lg(2ˣ) = lg 16
The law lg(aᵖ) = p · lg a brings x down.
x · lg 2 = lg 16
We divide both sides by lg 2.
x = lg 16 / lg 2 = 4
We put in x = 4: 3 · 2⁴ = 3 · 16 = 48. That is correct.
Pitfalls
Thinking that 3 · 2ˣ is the same as 6ˣ.
The power is worked out first, so 3 · 2ˣ is 3 times 2ˣ. Divide by the three before taking lg.
Cancelling lg 16 / lg 2 down to lg 8.
lg 16 / lg 2 = 4, but lg 8 ≈ 0.903. Logarithms cannot be cancelled like that.
Practise on your own
0 of 4 correct1Work out lg 10 000 without a calculator.
2Solve the equation 10ˣ = 40. Give your answer to two decimal places.
3Solve the equation 2 · 3ˣ = 54.
4Solve the equation 200 · 1.05ˣ = 300. Give your answer to one decimal place.
Common questions
What is the difference between lg and ln?
lg asks for the exponent when the base is 10. ln asks for the exponent when the base is the number e ≈ 2.718. The laws are the same.
Why can you not take lg of zero or a negative number?
10 to the power of any number is positive. No exponent gives 0 or −5, so there is no logarithm.
Are you allowed to take lg of both sides of an equation?
Yes, when both sides are positive. Equal numbers have equal logarithms, so the equality still holds.
Read also
- Maths 1–3Exponential functions: y = C · aˣWhat is an exponential function, and what do C and a mean?
- Years 7–9 and upper secondaryPowers and the laws of exponentsHow do you calculate with powers?
- Years 8–9 and upper secondaryChange factor (percentage multiplier)What is a change factor and how do you use it?
- Years 8–9 and upper secondaryInterest and compound interestHow do you work out interest and compound interest?
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