Logarithm Calculator
Compute logarithms with any base, natural and common logs, and solve exponential equations — results update as you type.
Log with any base
log10(100) =
2
Change of base: ln(100) / ln(10) = 4.605170186 / 2.302585093
Quick ln and log10
ln(x)
0.9162907319
log10(x)
0.3979400087
Solve bx = y for x
x = log2(64) =
6
Evaluate bx
210 =
1024
Log rules reference
| Product | logb(xy) = logb(x) + logb(y) |
| Quotient | logb(x/y) = logb(x) − logb(y) |
| Power | logb(xk) = k · logb(x) |
| Change of base | logb(x) = ln(x) / ln(b) |
| Identity | logb(b) = 1, logb(1) = 0 |
Results are rounded to 10 significant digits using double-precision arithmetic.
How it works
- Enter a base and an argument in the first panel to get log base b of x, with the change-of-base formula written out.
- Type a single number in the quick row to see its natural logarithm ln(x) and common logarithm log10(x) side by side.
- Use the inverse panel to solve b^x = y for the unknown exponent x, or evaluate b^x directly for any real exponent.
Frequently asked questions
- How do you calculate a logarithm with any base?
- The calculator uses the change-of-base formula: log base b of x equals ln(x) divided by ln(b). This works for every valid base because natural logarithms are defined for all positive numbers. The formula with your actual numbers is displayed under the result so you can follow the computation.
- Why can’t the argument of a logarithm be zero or negative?
- A logarithm asks which exponent turns the base into the argument, and a positive base raised to any real power is always positive. There is therefore no real exponent that produces zero or a negative number. When you enter such a value the calculator explains the domain error instead of returning NaN.
- Why is base 1 not allowed?
- Every power of 1 equals 1, so the equation 1^x = y has no solution unless y is exactly 1, and then every x works. Because a function must give one defined answer, log base 1 is undefined. Bases must be positive and different from 1; common choices are 2, e, and 10.
- How accurate are the results?
- Calculations use IEEE 754 double-precision floating point, the same standard used by scientific software, giving roughly 15-16 significant digits internally. Results are displayed rounded to 10 significant digits, which is more than enough for coursework, engineering estimates, and pH or decibel calculations.
- Is my data sent to a server?
- No. The entire calculator runs client-side in JavaScript, so nothing you type leaves your device and results appear instantly without any network round trip. It also keeps working offline once the page has loaded.
About this tool
This logarithm calculator answers the question a log always asks: to what power must the base be raised to produce a given number? Enter any valid base and argument and it returns log base b of x immediately, alongside quick readouts of the natural logarithm ln(x) and the common logarithm log10(x). A third panel works in reverse — it solves the exponential equation b^x = y for the unknown exponent, and evaluates b^x for any real exponent when you already know x.
Everything runs directly in your browser with double-precision floating-point arithmetic; no numbers are uploaded anywhere. Arbitrary bases are handled with the change-of-base identity log_b(x) = ln(x) / ln(b), and the calculator prints this formula with your values substituted so you can check the working or reproduce it on paper or in an exam. Invalid inputs — a zero or negative argument, a non-positive base, or base 1 — produce a plain-language explanation of why that logarithm is undefined rather than a cryptic error.
Logarithms appear anywhere quantities span many orders of magnitude. Chemists use log10 for pH, audio engineers use it for decibels, seismologists for earthquake magnitude, and programmers use log2 to reason about binary search steps, tree depths, and how many bits a number needs. The inverse solver is handy for growth problems: if an investment or population multiplies by b each period, solving b^x = y tells you how many periods reaching y will take.
A few practical tips: to get log2 quickly, set the base to 2 in the first panel. Results with long decimals are rounded to 10 significant digits, so tiny trailing digits you may see elsewhere are rounding artifacts, not errors. If you forget a rule mid-problem, open the log rules reference below the calculator — it lists the product, quotient, power, and change-of-base identities in one place.