Logarithm Calculator
Calculate logarithms with any base
Calculate log base b of x
log_b(x) = ln(x) / ln(b)What is a Logarithm Calculator?
A Logarithm Calculator is a math tool that computes logarithms, which answer the question: "What power do I raise a base to in order to get a number?" For example, if 10³ = 1000, then log₁₀(1000) = 3. The logarithm tells you the exponent (power) needed to turn 10 into 1000.
Logarithms are widely used in math and science because they help work with very large or very small numbers, convert multiplication into addition, and model real-world growth and decay. They appear in fields like chemistry (pH), finance (compound growth), engineering, computer science, and statistics.
This calculator computes all three major types of logarithms simultaneously:
Supported Logarithm Types
- Common logarithm (base 10) -- log₁₀(x), often written as log(x)
- Natural logarithm (base e) -- ln(x), where e ≈ 2.71828
- Custom base logarithm -- log₂(x) -- enter any valid base b to compute log_b(x)
How to Use This Logarithm Calculator
- Enter the number (x) -- the value you want to take the logarithm of
- Enter the base (b) -- defaults to 10, but you can change it to any valid base (e.g., 2, e, 5)
- Click "Calculate" -- to compute the logarithm
- Review all three results -- the calculator shows log_b(x), ln(x), and log₁₀(x) simultaneously
- Use the result -- apply it in your equation, problem, or real-world calculation
Tips:
- For real-number results, the input x must be greater than 0
- The base b must be greater than 0 and b ≠ 1
- If your result looks unexpected, double-check whether you need log (base 10) vs ln (base e)
Logarithm Formulas
Definition of a Logarithm
log_b(x) = y means bʸ = x
The logarithm returns the exponent y that makes bʸ equal to x
Common Logarithm
log₁₀(x)
Base 10, often written as log(x)
Natural Logarithm
ln(x)
Base e, where e ≈ 2.71828
Change of Base Formula
Compute any base using log base 10 or ln:
log_b(x) = ln(x) / ln(b) = log₁₀(x) / log₁₀(b)
Convert between any bases using this identity
Useful Logarithm Rules
Product Rule
log_b(xy) = log_b(x) + log_b(y)
Quotient Rule
log_b(x/y) = log_b(x) − log_b(y)
Power Rule
log_b(xᵏ) = k × log_b(x)
Log of 1 / Log of Base
log_b(1) = 0
log_b(b) = 1
Example Calculations
Example 1: Common Log (Base 10)
Compute: log₁₀(1000)
Reasoning: 10³ = 1000
Result: 3
Example 2: Natural Log (Base e)
Compute: ln(e²)
Reasoning: ln returns the exponent when the base is e
Result: 2
Example 3: Custom Base Log
Compute: log₂(32)
Reasoning: 2⁵ = 32
Result: 5
Example 4: Using the Change of Base Formula
Compute: log₅(125)
Direct reasoning: 5³ = 125, so log₅(125) = 3
Change of base: ln(125) / ln(5) = 4.8283 / 1.6094 = 3
Result: 3
Frequently Asked Questions
What's the difference between log and ln?
log(x) usually means base 10 (common log), while ln(x) means base e (natural log). They're both logarithms—just with different bases.
Why can't I take the logarithm of 0 or a negative number?
In real-number math, log values are only defined for x > 0. There is no real exponent that makes a positive base equal 0 or a negative number.
What base values are allowed?
The base must be greater than 0 and not equal to 1. A base of 1 would always equal 1 for any exponent, so it can't produce different outputs.
What does a logarithm output represent?
The output is the exponent. If log_b(x) = y, then bʸ = x. That's the core meaning of logarithms.
When are logarithms useful in real life?
Logs are used when quantities change multiplicatively or span wide ranges: pH in chemistry, earthquake magnitude scales, sound intensity (decibels), compound growth/interest, and many scientific models.
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