A logarithm is the inverse of exponentiation: log_b(x) = y means bʸ = x. So log₂(8) = 3 because 2³ = 8. Logarithms answer the question "what exponent do I need?"

What's the difference between log and ln?

log usually means the common logarithm (base 10), while ln is the natural logarithm (base e ≈ 2.71828). Base 10 suits orders of magnitude (pH, decibels, Richter scale); base e arises naturally in calculus, growth, and decay problems.

How do I calculate a log with an unusual base?

Use the change-of-base formula: log_b(x) = ln(x) ÷ ln(b) (any common base works). For example, log₇(50) = ln 50 ÷ ln 7 ≈ 2.011. This is how calculators evaluate arbitrary bases internally.

What are the basic logarithm rules?

Product: log(xy) = log x + log y. Quotient: log(x/y) = log x − log y. Power: log(xⁿ) = n·log x. These rules turn multiplication into addition and powers into multiplication, which is the historical reason logs were invented.

Why can't I take the log of zero or a negative number?

Because no real exponent makes a positive base produce zero or a negative result — bʸ is always positive. Logarithms of non-positive numbers are undefined in the real numbers (complex analysis extends them, but standard calculators return an error).

Log Calculator

Calculate logarithms with various bases including natural and common logs. Free, fast, accurate — no signup, mobile-frie

Calculation Type

Value (x)

Number to find logarithm of

Base (b)

Base of logarithm (must be > 0 and ≠ 1)

📐 Common Logarithms

Logarithm Properties

log₍b₎(xy) = log₍b₎(x) + log₍b₎(y)

log₍b₎(x/y) = log₍b₎(x) - log₍b₎(y)

log₍b₎(x^n) = n·log₍b₎(x)

log₍b₎(b) = 1

log₍b₎(1) = 0

Main Result

log₍10₎(100)

Natural Log

4.60517

ln(x) = log₍ₑ₎(x)

Common Log

log₁₀(x)

Binary Log

6.643856

log₂(x)

Antilog (base 10)

1.000e+100

10^100

Base 2 Result

6.643856

log₍2₎(100)

Different Notations

Exponential:

2.0000e+0

Scientific:

2.00000

Engineering:

2.000 × 10^0

Function Properties

Domain: (0, ∞)

Range: (-∞, ∞)

Vertical asymptote: x = 0 (vertical)

Inverse function: y = 10^x

Step-by-Step Solution

1Calculate: log₍10₎(100)

2Using change of base formula: log₍10₎(100) = ln(100) / ln(10)

3ln(100) = 4.605170

4ln(10) = 2.302585

5Result: 4.605170 / 2.302585 = 2.000000

6Verification: 10^2.000000 ≈ 100.00

How it works

A logarithm answers “what power do I raise the base to, to get this number?” It's the inverse of exponentiation. Common bases are 10 (log) and e (natural log, ln). Logs turn multiplication into addition, which is why they appear in scales like decibels and pH.

Logarithm

logᵦ(x) = y   means   bʸ = x

b: the base
x: the number
y: the exponent (the log)

Worked example

Compute log₁₀(1000)

What power of 10 gives 1000?
10³ = 1000

log₁₀(1000) = 3.

Good to know

log(ab) = log a + log b — turning products into sums is the whole point of logarithms.
Change of base: logᵦ(x) = ln(x) ÷ ln(b), so any calculator with ln can do any base.
You can't take the log of zero or a negative number.