Logarithm Calculator

This calculator computes logarithms with any base online using our free calculator.

How the Logarithm Calculator Works

log_b(x) = y means b^y = x | Change of base: log_b(x) = ln(x) / ln(b)

Logarithm is the inverse of exponentiation. If b^y = x, then log_b(x) = y. Common logarithm (log) uses base 10, natural logarithm (ln) uses base e (approximately 2.718). Logarithms are used in measuring earthquake intensity, sound levels, and algorithm complexity.

1
Select Logarithm Type
Choose base 10 (log), natural (ln), base 2, or custom base
2
Enter Value
Input the positive number to calculate its logarithm
3
Enter Custom Base
If using custom base, enter a positive value not equal to 1
4
View Result
The result displays with the calculation method explained

Use Cases

Earthquake Measurement

The Richter scale uses common logarithm to measure earthquake magnitude

Sound Intensity

Decibel measurements for sound levels are logarithmic

Exponential Growth

Calculate half-life periods, population growth, and radioactive decay

Computer Science

Analyze algorithm complexity (O(log n)) and data compression

Tips

1
log(1) = 0 for any base (because b^0 = 1)
2
log_b(b) = 1 always (because b^1 = b)
3
log(a x b) = log(a) + log(b)
4
log(a/b) = log(a) - log(b)
5
log(a^n) = n x log(a)

Common Mistakes

Calculating logarithm of zero or negative numbers - log is only defined for positive values
Confusing log and ln - log usually means base 10, ln means base e
Assuming log(a+b) = log(a) + log(b) - this is incorrect; the rule is log(a x b) = log(a) + log(b)
Using base 1 or negative base - base must be positive and not equal to 1

FAQs

What is the difference between log and ln?

log (common logarithm) has base 10, while ln (natural logarithm) has base e (approximately 2.718).

How do I calculate logarithm with a different base?

Use change of base: log_b(x) = log(x) / log(b) = ln(x) / ln(b). Example: log_2(8) = 3.

What are the basic properties of logarithms?

1) log(ab) = log(a) + log(b), 2) log(a/b) = log(a) - log(b), 3) log(a^n) = n log(a), 4) log(1) = 0.

Why is logarithm undefined for negative numbers?

Because any positive base raised to any real power gives a positive result. There is no real power that makes 10^x negative.

What is e and why is it important?

e is approximately 2.71828 and is the base of natural logarithm. It appears in continuous growth, compound interest, and many natural phenomena.

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Advanced Logarithm Calculator

Calculate logarithm with any base with advanced options.

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