Logarithm Calculator
Logarithm Calculator
Enter a number and a base to find its logarithm. Use base 10 for the common logarithm, or enter 2.71828 (e) for the natural logarithm — this works for any positive base other than 1.
A logarithm answers the question "this base raised to what power gives this number?" — it's the direct inverse of exponentiation, the same way subtraction inverts addition or division inverts multiplication. The number itself must be positive and the base must be positive and not equal to 1, since raising 1 to any power always gives 1 and never anything else — logarithms of zero, negative numbers, or with an invalid base are simply undefined. Base-10 (common) and base-e (natural) logarithms are by far the most frequently used in science, engineering, and finance, which is why they get dedicated named functions on most calculators.
- Definition: log base b of x asks "b to what power equals x?" — it's the inverse operation of exponentiation.
- Domain restrictions: the number (x) must be positive, and the base must be positive and not equal to 1 — logarithms of zero, negative numbers, or with an invalid base are undefined.
- Common bases: base 10 is the "common logarithm" (widely used in science and engineering), while base e ≈ 2.71828 is the "natural logarithm," commonly written ln(x) and used throughout calculus and growth/decay modeling.
How do I calculate the natural logarithm (ln)?
Enter 2.718281828 (or more decimal places of e) as the base — the natural logarithm is simply log base e, so this calculator handles it the same way as any other base.
Why is the logarithm of a negative number undefined?
No real exponent applied to a positive base can ever produce a negative result, so there's no real value that satisfies the equation for a negative input — the logarithm only extends to complex numbers in more advanced math.
Logarithm Calculator
Enter a number and a base to find its logarithm. Use base 10 for the common logarithm, or enter 2.71828 (e) for the natural logarithm — this works for any positive base other than 1.
A logarithm answers the question "this base raised to what power gives this number?" — it's the direct inverse of exponentiation, the same way subtraction inverts addition or division inverts multiplication. The number itself must be positive and the base must be positive and not equal to 1, since raising 1 to any power always gives 1 and never anything else — logarithms of zero, negative numbers, or with an invalid base are simply undefined. Base-10 (common) and base-e (natural) logarithms are by far the most frequently used in science, engineering, and finance, which is why they get dedicated named functions on most calculators.
- Definition: log base b of x asks "b to what power equals x?" — it's the inverse operation of exponentiation.
- Domain restrictions: the number (x) must be positive, and the base must be positive and not equal to 1 — logarithms of zero, negative numbers, or with an invalid base are undefined.
- Common bases: base 10 is the "common logarithm" (widely used in science and engineering), while base e ≈ 2.71828 is the "natural logarithm," commonly written ln(x) and used throughout calculus and growth/decay modeling.
How do I calculate the natural logarithm (ln)?
Enter 2.718281828 (or more decimal places of e) as the base — the natural logarithm is simply log base e, so this calculator handles it the same way as any other base.
Why is the logarithm of a negative number undefined?
No real exponent applied to a positive base can ever produce a negative result, so there's no real value that satisfies the equation for a negative input — the logarithm only extends to complex numbers in more advanced math.