Why scientists love shrinking big numbers
The Richter scale, pH values, and decibels all share one trick: they use logarithms to squeeze enormous ranges into small, manageable numbers. A logarithm answers the question: to what power must I raise the base to get this number? That one question powers the log calculator on this page.
The logarithm relationship
Formula: If log_b(x) = y, then b^y = x.
In words: the log of x in base b is the exponent y you would need on b to produce x. The most common bases are 10 (common log, used in pH and decibels), e (natural log, used in growth and decay), and 2 (used in computing and information theory).
Using the calculator
- Enter the number you want the logarithm of.
- Enter the base, or pick base 10, e, or 2.
- Read the result and check it by raising the base to that power.
A worked example, step by step
Find log_10(1000).
- Step 1: ask what power of 10 gives 1000.
- Step 2: 10^1 = 10, 10^2 = 100, 10^3 = 1000.
- Step 3: the exponent is 3, so log_10(1000) = 3.
The answer is 3. As a second check, log_2(32) = 5 because 2^5 = 32.
Where logs show up
They describe earthquake magnitude, sound loudness, acidity, algorithm complexity, and the time needed for investments to grow. To undo a logarithm, use the antilog calculator; for the center of a dataset, see the standard deviation calculator.
Mistakes that give the wrong answer
- Taking the log of zero or a negative number. Real logarithms are undefined there; the calculator will refuse rather than guess.
- Using the wrong base. Log base 10 and natural log give very different answers for the same input, so confirm which one your textbook or instrument expects.
- Splitting sums. Log(a + b) is not log(a) + log(b). Only multiplication splits: log(a x b) = log(a) + log(b).
Finding logs on a physical calculator
Handheld calculators usually offer LOG for base 10 and LN for base e, with other bases reached through the change-of-base rule: log_b(x) = log_10(x) / log_10(b). To find log_2(50), divide log_10(50) by log_10(2): 1.699 / 0.301 = 5.64. As a sanity check, 2^5 = 32 and 2^6 = 64, so 5.64 sits exactly where it should. One more habit worth building: after computing a logarithm, raise the base to the result and confirm you get your input back. That ten-second verification catches wrong-base and typo errors before they spread into the rest of your work.
What this assumes
Results are real numbers. The base must be positive and different from 1, and the input must be positive. When in doubt, recheck the base first: most logarithm errors trace back to a wrong base, not wrong arithmetic.