Logarithm Calculator

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x: 256 base: 2 precision: 6 value: 8 raw: 8 show steps: 1

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Online logarithm calculator

The logarithm log_b(x) = y means b^y = x. The calculator calculates log10, ln (base e), log2 and any log_b(x) and shows the formulas and properties of logarithms needed for the exam.

logarithm calculator ln log10 log2 natural logarithm formulas logarithms

Definition and types

log10(x): decimal logarithm (Richter scale, pH, decibels). ln(x): natural logarithm – base e=2.71828 (exponential growth, integrals). log2(x): binary logarithm (computer science, entropy, bits).

Properties of

log(a×b) = log(a) + log(b). log(a/b) = log(a) - log(b). log(a^n) = n×log(a). Change of basis: log_b(x) = ln(x)/ln(b). These properties simplify calculations for very large and small numbers.

Examples - the most common logarithms

Expression	Result	Why
log₁₀(100)	2	10² = 100
log₁₀(1000)	3	10³ = 1000
log₂(8)	3	2³ = 8
log₂(1024)	10	2¹⁰ = 1024
ln(e)	1	e¹ = e ≈ 2.718
log₅(125)	3	5³ = 125
log₃(81)	4	3⁴ = 81
log₁₀(1)	0	each base⁰ = 1

How to calculate logarithm without a calculator?

In school tasks, logarithm can almost always be reduced to the question "to what power should the base be raised?". Factoring helps: log₂(32) → 32 = 2⁵ → result 5. When a number is not a nice power, use the change of base formula: log_b(x) = log(x) ÷ log(b). Remember about the domain: the logarithm only exists for x > 0, with base b > 0 and b ≠ 1.

Where will you encounter logarithms in practice

The pH scale (pH = −log[H⁺]), decibels in acoustics (10·log power ratio), the Richter scale for earthquakes, the brightness of stars, and in computer science - the complexity of algorithms O(log n) and the number of bits needed to store the number (log₂). Wherever values increase exponentially, the logarithm "straightens" the scale to readable numbers.

Frequently asked questions

Where do we use logarithms in practice?

Richter Scale (earthquakes), pH of acids, decibels (sound), investment doubling time (rule of 72), Big O in algorithms (O log n), compression (Shannon entropy).

What is the difference between log and ln?

log (without base) = log10 in school mathematics and engineering. ln = log_e (natural logarithm). In computer science, log often means log2.

How to calculate logarithm to any base?

log_b(x) = log10(x) / log10(b) = ln(x) / ln(b). E.g. log_3(81) = ln(81)/ln(3) = 4.38/1.10 = 4. (Because 3^4=81.)

Other mathematical calculators: GCA/GCD and power calculator.

You can also calculate the inverse operation of exponentiation inradical calculator.