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    Logarithm Calculator

    The logarithm of a number to any base — ln, log and log₂: exact when the number is a power of the base, with the check and the graph

    Logarithm Calculator

    Log base
    x

    Natural, Common and Binary Logarithms

    logb x is the power you raise b to, to get x: by = x; b > 0, b ≠ 1, x > 0.

    ln is the natural log, base e ≈ 2.71828; log is the common log, base 10; log₂ is the binary log, base 2.

    Table of logarithms
    xln xlog xlog₂ x
    1000
    2≈ 0.6931≈ 0.30101
    3≈ 1.0986≈ 0.4771≈ 1.5850
    4≈ 1.3863≈ 0.60212
    5≈ 1.6094≈ 0.6990≈ 2.3219
    6≈ 1.7918≈ 0.7782≈ 2.5850
    7≈ 1.9459≈ 0.8451≈ 2.8074
    8≈ 2.0794≈ 0.90313
    9≈ 2.1972≈ 0.9542≈ 3.1699
    10≈ 2.30261≈ 3.3219

    Example: ln 2

    ln 2 ≈ 0.6931

    ln 2 is the power you raise e ≈ 2.71828 to, to get 2: ey = 2.

    e⁰ = 1 < 2 < e¹ ≈ 2.7183, so 0 < ln 2 < 1

    Oxy1

    Answer point ≈ (2, 0.6931).

    The curve y = ln x passes through (1, 0) and (e, 1) and rises: for x > 1 the log is positive, for 0 < x < 1 negative.

    Common mistakes

    • log without a base: on a calculator and at school it is base 10 (log 100 = 2), while Python's math.log and Excel's LN use base e (ln 100 ≈ 4.6052); Excel's LOG is base 10 again.
    • ln 0 and the log of a negative number do not exist: e raised to any power is positive.
    • There is no base 1: 1 raised to any power is 1.

    See Also

    Last Results

    FAQ

    What is the difference between ln, log and lg?

    ln is the natural log, base e ≈ 2.71828; log on a calculator is the common log, base 10 (some textbooks write it lg). They differ by a constant factor: log x = ln x / ln 10, so ln 2 ≈ 0.6931 and log 2 ≈ 0.3010.

    How to calculate log base 2 on a calculator?

    Divide two natural logs: log₂ 5 = ln 5 / ln 2 ≈ 1.609438 / 0.693147 ≈ 2.3219. Any base works the same way: log_b x = ln x / ln b.

    What is ln of infinity?

    ln x has no maximum: it grows without bound as x grows, only slowly — ln 1,000,000,000 ≈ 20.7233. So ln ∞ is not a number; one says that ln x tends to infinity.