Logarithm Calculator (log, ln, log2) and Exponents

This logarithm calculator finds the logarithm of a number in any base (10, e, 2, or one you choose) and does the reverse operation too: the base raised to a power, including e to the x and 10 to the x.

Whichever you ask for, the cards also show the common, natural and binary logarithms of the same number and the three common powers of it.

A logarithm answers the question "what power?". The base 10 logarithm of 1,000 is 3 because 10³ is 1,000. The base 2 logarithm of 1,024 is 10 because ten doublings make 1,024. Logarithms turn multiplication into addition and powers into multiplication, which is why they appear wherever quantities grow or shrink by a constant factor: decibels, the pH scale, earthquake magnitudes, compound growth and the number of bits a value needs.

The NIST Digital Library of Mathematical Functions defines the logarithm in any base as the natural logarithm divided by the natural logarithm of the base, and a general power as e raised to the exponent times the natural log of the base. The calculator works from those definitions to 34 significant digits.

1. Choose Logarithm to find log base b of x, or Power to find b to the power x.
2. Enter x.
3. Choose the base: 10, e, 2, or another base, which you then type in the box below.
4. Read the result at the top and the sentence under it. The cards give log₁₀ x, ln x, log₂ x, eˣ, 10ˣ and 2ˣ for the same x.

1. Natural logarithm. ln x is computed by decimal.js for x greater than zero. The logarithm of zero or a negative number is not a real number, and the card says so.

1. Logarithm in another base. The change of base rule, DLMF 4.2.8, divides the natural logarithm of x by the natural logarithm of the base. The base must be positive and not 1. When x is an exact power of the base, the result is exact: log₁₀ 1000 is 3, not 2.9999999999.

1. Powers. b to the power x is computed as e raised to x ln b (DLMF 4.2.28). A negative base with a non-whole exponent has no real value, and 0 to a negative power is undefined; the calculator reports both instead of returning a number.

1. e. The base e is computed as e¹ to 34 digits, 2.718281828459045..., not typed in.

1. Show the result. Results are rounded half up to 15 significant figures, with thousands separators, and switch to scientific notation at 10¹⁵ and above, so e to the 1,000 appears as 1.97007111401705 × 10⁴³⁴.

Using the defaults, log base 10 of 1,000:

1. 1,000 = 10 × 10 × 10 = 10³.
2. So log₁₀ 1000 = 3.
3. ln 1000 = 6.90775527898214, and log₂ 1000 = 9.96578428466209.
4. Check the change of base: 6.90775527898214 ÷ ln 10 (2.30258509299405) = 3.

A base the calculator buttons rarely have, from the log₅ 30 preset: ln 30 ÷ ln 5 = 2.11328275255938, and 5 to that power is 30.

The natural logarithm, written ln, uses the base e, about 2.71828. It is "natural" because the rate of growth of eˣ is eˣ itself, which makes e the base that falls out of calculus, continuous compounding and radioactive decay. The ln 10 preset gives ln 10 = 2.30258509299405.

Natural and common logarithms differ only by that constant factor: ln x = 2.30258509299405 × log₁₀ x.

Switch to power mode and choose base e or 10, or read the eˣ and 10ˣ cards in either mode. The e to the 2 preset gives e² = 7.38905609893065. Powers of 10 are exact: 10 to the power 3 is 1,000, and 10 to the power 2 is 100.

Powers are also how exponential change works. Something that grows 5% per period is multiplied by 1.05 each time, so after 10 periods it has grown by a factor of 1.05¹⁰. The 1.05 to the 10 preset gives 1.62889462677744. A logarithm runs that backwards: the number of periods needed to reach a factor F is log base 1.05 of F.

The reference table gives all three logarithms for numbers from 0.001 to 1,000,000. Notice the patterns: logarithms of numbers below 1 are negative, the logarithm of 1 is 0 in every base, and log₂ of a power of two (2, 4, 8, 16, 64, 1,024) is a whole number.

• Complex logarithms and powers. The logarithm of a negative number, and a negative number to a fractional power, are complex. Only real results are given. Powers of i, which cycle through i, -1, -i and 1, are outside the real numbers this calculator handles.
• Logarithm of zero. It has no value; it tends to minus infinity as x approaches 0.
• Extremely large exponents. Results beyond about 10 to the power 9 quadrillion overflow decimal.js and are reported as not a real number.
• Precision of the base e. e is irrational; 34 significant digits are carried and 15 shown.

• Assuming log means the same base everywhere. In most school maths and on calculators log means log₁₀; in many programming languages and higher maths, log means ln. Check which one your source uses.
• Splitting the log of a sum. log(a + b) is not log a + log b. The rule is for products: log(ab) = log a + log b.
• Taking the log of zero or a negative number. It is undefined in the real numbers.
• Reading log₂ as log × 2. The 2 is the base, not a multiplier. "2 log 2" means 2 times log 2.
• Confusing eˣ with 10ˣ. e² is 7.38905609893065; 10² is 100.

• NIST Digital Library of Mathematical Functions, section 4.2, Definitions: the natural logarithm, logarithms to a general base (4.2.8) and general powers (4.2.28). dlmf.nist.gov/4.2
• OpenStax, Algebra and Trigonometry 2e, section 6.3, Logarithmic Functions: the logarithm as the inverse of the exponential, common and natural logarithms. openstax.org/books/algebra-and-trigonometry-2e/pages/6-3-logarithmic-functions