Use the free Logarithm Calculator below: your data never leaves your browser — all work happens on your device, and nothing is uploaded to any server.
What This Calculator Does
A logarithm calculator answers the question: to what power must I raise the base to get this number? If you know that 10 to the power of 2 is 100, then the logarithm of 100 with base 10 is 2. Written as log base 10 of 100 equals 2. Logarithms turn multiplication into addition and turn huge ranges into small, manageable numbers. That is why they appear in science, finance, music, and computer science, from earthquake magnitudes to sound levels to the time complexity of algorithms.
This tool computes a logarithm of any base, not just base 10. It also shows its working in three steps, using the change-of-base rule: the log of x with base b equals the natural log of x divided by the natural log of b. The natural log uses base e, about 2.718. Because every scientific calculator has a natural log button, this rule lets you compute any logarithm from one button. The tool also checks its answer by raising the base to the result, so you can see the calculation close the loop.
How to Use It
Type the number x, then type the base b. The default base is 10, the common logarithm used in science and engineering. For natural logarithms, set the base to 2.718281828 or simply use base e in your head and read the step values. Click Calculate. You will see the result to six decimal places, the two natural logs used, and the check. Remember the rules: x must be greater than zero, and the base must be positive and not equal to 1. Logarithms of zero or negative numbers are not real numbers.
The Change-of-Base Rule, Explained
Suppose you want log base 2 of 8. You know the answer is 3, because 2 cubed is 8. The change-of-base rule says: take the natural log of 8, which is about 2.079, and divide it by the natural log of 2, which is about 0.693. The division gives 3. The rule works because logarithms with different bases are proportional to each other: changing the base only multiplies every logarithm by a constant. That constant is exactly one divided by the log of the new base. The tool shows both natural logs so you can follow each division by hand.
A Worked Example
Find log base 10 of 1000. The natural log of 1000 is about 6.908, and the natural log of 10 is about 2.303. Divide 6.908 by 2.303 and you get 3, which matches the fact that 10 cubed is 1000. Now try log base 2 of 100. The natural log of 100 is about 4.605, and the natural log of 2 is about 0.693. Divide and you get about 6.644. Check: 2 to the power of 6.644 is about 100. Notice that a logarithm does not have to be a whole number. Most are not, and the decimal part carries real meaning.
Common Mistakes
The most common mistake is trying to take the log of a negative number or zero and expecting a real answer. There is none, though advanced mathematics uses complex numbers for that case. Another mistake is confusing the base with the number: log base 2 of 8 is 3, but log base 8 of 2 is one third, because 8 to the power of one third is 2. A third mistake is forgetting that log base 10 of 100 is 2, while the natural log of 100 is about 4.605. Always check which base your problem uses before you start.
Frequently Asked Questions
What is log base e called? The natural logarithm, written ln. It is the inverse of the exponential function e to the power of x. What is log base 2 used for? It measures doubling: how many times you must double 1 to reach a number. Computer scientists use it for binary search and information. Why do scientists use logs? Because they compress wide ranges. Earthquake energy, sound intensity, and acidity all span many orders of magnitude, and a log scale keeps them readable.