Free Logarithm Calculator with Steps
Calculate common log, natural log (ln), and antilogarithms with step-by-step working. Perfect for A-Level and GCSE maths.
Step-by-step learning with explanations
Choose operation
Find what power the base must be raised to get the input
Select base type
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Quick Examples
- •log₁₀(x) asks: "10 to what power equals x?"
- •ln(x) uses base e ≈ 2.71828
- •Antilog reverses the operation
- •Can't take log of 0 or negative numbers
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.
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
The natural log 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).
Essential Log Laws
Product Rule
log(ab) = log(a) + log(b)
The log of a product equals the sum of logs
Quotient Rule
log(a/b) = log(a) - log(b)
The log of a quotient equals the difference of logs
Power Rule
log(aⁿ) = n × log(a)
The log of a power brings the exponent down
Change of Base
logb(x) = ln(x)/ln(b)
Convert any log using natural logs
Worked Examples
Example 1: Calculate log₁₀(1000)
Step 1: Ask "10 to what power = 1000?"
10¹ = 10, 10² = 100, 10³ = 1000
Step 2: We found 10³ = 1000
Answer: log₁₀(1000) = 3
Example 2: Calculate log₂(32)
Step 1: Use change of base formula
log₂(32) = ln(32) / ln(2)
Step 2: Calculate
= 3.466 / 0.693 = 5
Verify: 2⁵ = 32 ✓
Answer: log₂(32) = 5
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.
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