Free Logarithm Calculator with Steps
Calculate common log, natural log (ln), and antilogarithms with step-by-step working and interactive visualisations. Perfect for A-Level and GCSE maths.
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What is a Logarithm?
A logarithm is the inverse operation of exponentiation. It answers the question: "What power must we raise the base to in order to get a certain number?"
For example, log₁₀(100) = 2 because 10² = 100. We're asking: "10 to what power equals 100?" The answer is 2.
In A-Level maths, logarithms appear in the "Exponentials and Logarithms" unit and are essential for solving exponential equations, growth and decay models, and integration.
The Definition
If by = x, then logb(x) = y
Types of Logarithms
Common Logarithm (log)
Base 10 — written as log(x) or log₁₀(x)
log(100) = 2 (because 10² = 100)
log(1000) = 3 (because 10³ = 1000)
Natural Logarithm (ln)
Base e ≈ 2.71828 — used heavily in calculus
ln(e) = 1 (because e¹ = e)
ln(e²) = 2 (power rule)
Custom Base
Any positive base except 1 — common in computer science
log₂(8) = 3 (because 2³ = 8)
log₅(25) = 2 (because 5² = 25)
💡 A-Level Tip
ln(x) is preferred in calculus because d/dx[ln(x)] = 1/x. For exponential equations like 2ˣ = 8, take ln of both sides: x·ln(2) = ln(8).
The 6 Logarithm Laws
These six laws are fundamental to A-Level maths. They let you simplify, expand, and solve logarithmic expressions.
Product Rule
log_b(AB) = log_b(A) + log_b(B)
Follows from b^m · b^n = b^(m+n)
log₂(8) = log₂(2×4) = log₂(2) + log₂(4) = 1 + 2 = 3
Quotient Rule
log_b(A/B) = log_b(A) − log_b(B)
Follows from b^m ÷ b^n = b^(m−n)
log₂(4) = log₂(8/2) = log₂(8) − log₂(2) = 3 − 1 = 2
Power Rule
log_b(xⁿ) = n · log_b(x)
Follows from (b^m)^n = b^(mn)
log₂(8) = log₂(2³) = 3 × log₂(2) = 3 × 1 = 3
Change of Base
log_b(x) = ln(x) / ln(b)
Let y = log_b(x), then b^y = x, take ln both sides
log₂(100) = ln(100)/ln(2) ≈ 4.605/0.693 ≈ 6.644
Log of 1
log_b(1) = 0
Because b⁰ = 1 for any base b
log₁₀(1) = 0, log₂(1) = 0, ln(1) = 0
Log of Base
log_b(b) = 1
Because b¹ = b for any base b
log₁₀(10) = 1, log₂(2) = 1, ln(e) = 1
Change of Base Formula
The change of base formula allows you to compute any logarithm using only the buttons available on a standard calculator (log₁₀ or ln).
Most scientific calculators have two logarithm buttons: log (base 10) and ln (base e). To compute log₂(100), for example, neither button directly works — so you use:
log_b(x) = ln(x) / ln(b)
Worked Example: log₂(100)
Step 1: Apply the formula
log₂(100) = ln(100) / ln(2)
Step 2: Compute each ln
= 4.60517 / 0.69315
Step 3: Divide
≈ 6.6439
Verify: 2^6.6439 ≈ 100 ✓
Domain, Range & Graph Behaviour
Domain: x > 0
You cannot take the log of 0 or a negative number. No real power of a positive base produces zero or a negative result.
Range: all real numbers
The output of a logarithm can be any real number — positive, negative, or zero. As x → 0⁺, log_b(x) → −∞. As x → ∞, log_b(x) → +∞.
Graph shape depends on base
Base constraints
• b > 0 — base must be positive
• b ≠ 1 — if b = 1, log_b is undefined (all powers of 1 equal 1)
Common Mistakes to Avoid
✗ log(a + b) ≠ log(a) + log(b)
The product rule only applies to multiplication: log(ab) = log(a) + log(b). There is no rule for addition inside a log.
✗ log(0) is undefined
The domain of log is x > 0. You cannot take the log of zero — it would approach −∞ but never be defined at exactly 0.
✗ Mixing up log₁₀ and ln
"log" usually means base 10 (common log). "ln" always means base e. These give different results: log(10) = 1, but ln(10) ≈ 2.303.
✗ log_b(b^n) = n, not b^n
The logarithm cancels the base: log_b(b^n) = n × log_b(b) = n × 1 = n. A common error is confusing the result with the original b^n.
✗ Forgetting log and antilog are inverses
b^(log_b(x)) = x and log_b(b^x) = x always. Antilog "undoes" log and vice versa — just like squaring and square root.
✗ log(a × b) ≠ log(a) × log(b)
The product rule says log(ab) = log(a) + log(b), not a product. The logs of individual factors are added, not multiplied.
Antilogarithms Explained
An antilogarithm (or antilog) is the inverse operation of a logarithm. If log_b(x) = y, then antilog_b(y) = b^y = x.
In other words: antilog raises the base to the given power. The antilog of 3 in base 10 is 10³ = 1000.
On a calculator, antilog of y in base 10 is the 10^x button. Antilog in base e uses the e^x or exp button.
Antilog Table (Base 10)
| y (log value) | antilog₁₀(y) = 10^y |
|---|---|
| 0 | 1 |
| 1 | 10 |
| 2 | 100 |
| 3 | 1000 |
| 0.5 | ≈ 3.162 |
| -1 | 0.1 |
Logarithms in Real Life
pH Scale (Chemistry)
pH = −log₁₀[H⁺]. Each unit change represents a 10× change in acidity.
Decibels (Sound)
dB = 10 × log₁₀(I/I₀). A 10 dB increase sounds twice as loud but is 10× the intensity.
Richter Scale (Earthquakes)
Each 1-point increase on the Richter scale is 10× more ground motion.
Binary Data (Computer Science)
log₂(n) bits are needed to represent n values. 8 bits = 2⁸ = 256 values.
Compound Interest (Finance)
Time to double = log(2) / log(1 + r) where r is the interest rate.
Worked Examples
Example: log₁₀(1000)
Ask: "10 to what power = 1000?"
10¹ = 10, 10² = 100, 10³ = 1000
Found 10³ = 1000
log₁₀(1000) = 3
Example: log₂(32)
Use change of base: ln(32) / ln(2)
= 3.4657 / 0.6931
= 5.0 (exactly)
Verify: 2⁵ = 32 ✓
log₂(32) = 5
Example: ln(e³)
Apply power rule: 3 × ln(e)
ln(e) = 1 by definition
= 3 × 1 = 3
ln(e³) = 3
Example: 10³ (Antilog)
Antilog₁₀(3) = 10³
= 10 × 10 × 10
= 1000
antilog₁₀(3) = 1000
Example: Solve 2ˣ = 50
Take log of both sides
x × log(2) = log(50)
x = log(50) / log(2)
x = 1.6990 / 0.3010
x ≈ 5.644
Example: log₃(1/9)
1/9 = 3⁻²
log₃(3⁻²) = −2 × log₃(3)
= −2 × 1
log₃(1/9) = −2
Frequently Asked Questions
What is a logarithm?
A logarithm answers: "What power must we raise the base to in order to get a certain number?" For example, log₁₀(100) = 2 because 10² = 100.
What is the difference between log and ln?
log uses base 10 (common logarithm), while ln uses base e ≈ 2.718 (natural logarithm). ln is preferred in calculus.
What is an antilogarithm?
An antilog is the inverse of a log. If log₁₀(x) = 2, then antilog₁₀(2) = 10² = 100. It "undoes" the logarithm.
Why can't I take log of a negative number?
No positive base raised to any real power can produce a negative number. That's why log(x) is undefined for x ≤ 0.
What is the change of base formula?
logₐ(x) = ln(x)/ln(a) = log(x)/log(a). This lets you calculate any log using just ln or log buttons.
Is this aligned with A-Level?
Yes! It covers all log laws, change of base, and solving exponential equations as required by Edexcel, AQA, and OCR.
What is the domain of a logarithm?
Domain is x > 0 (all positive reals). The base b must also be positive and not equal to 1. The range is all real numbers.
How are logarithms used in real life?
pH scale, decibels, Richter scale, binary data storage (log₂), and compound interest doubling time all use logarithms.
What is log_b(1)?
log_b(1) = 0 for any base b, because b⁰ = 1 always. This is one of the fundamental log identities.
What is log_b(b)?
log_b(b) = 1 for any base b, because b¹ = b. For example, log₁₀(10) = 1 and ln(e) = 1.
Can the base of a logarithm be a fraction?
Yes! As long as 0 < b < 1, you can use it as a base. log_{0.5}(0.25) = 2 because 0.5² = 0.25. The graph decreases instead of increasing.
What is the inverse of ln(x)?
The inverse of ln(x) is eˣ (also written exp(x)). These are inverse functions: e^(ln(x)) = x and ln(eˣ) = x.
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