Log Calculator: Logarithms, Natural Log and Change of Base

Logs get taught as a rule to memorise and then quietly turn up everywhere: compound growth, sound levels, acidity, the number of bits a value needs. The idea underneath is simpler than the notation makes it look.

What the notation is asking

Exponents and logs are the same fact read in opposite directions. Because 10 to the power 3 is 1,000, log₁₀(1,000) is 3. Because 2 to the power 6 is 64, log₂(64) is 6. The base is the number being multiplied by itself, and the log is how many times.

Most values are not tidy powers, which is why you need a calculator at all. log₁₀(500) is about 2.699, meaning 10 raised to 2.699 gets you to 500. It sits between 2 and 3, closer to 3, which matches the fact that 500 is closer to 1,000 than to 100 on a multiplying scale even though it is exactly halfway on an adding one.

The three bases you will actually meet

Base 10 when a log is written with no base at all, base e for anything growing continuously, and base 2 whenever you are counting doublings or bits.

Natural log, and why money uses it

Natural log uses base e, about 2.71828. It shows up wherever something grows continuously. To find how long money takes to double at 6% a year, divide ln(2) by ln(1.06): 0.6931 over 0.0583 gives 11.9 years. The old rule of 72 estimates 12 years for the same case, which is close enough for mental arithmetic and slightly off for anything else. The compound interest arithmetic behind that is the same formula rearranged.

Change of base

If your calculator only offers log and ln, you can still get any base you like. log_b(x) = ln(x) divided by ln(b). So log₂(1,000) is 6.9078 divided by 0.6931, which is 9.97. That tells you a value up to 1,000 needs 10 binary digits, since 2 to the power 9 is 512 and 2 to the power 10 is 1,024.

Where logs turn up in real numbers

pH is a log scale: pH equals minus log₁₀ of the hydrogen ion concentration. A concentration of 0.00001 mol per litre is 10 to the power minus 5, so the pH is 5. Move to pH 4 and the solution is not slightly more acidic, it is ten times more acidic.

Decibels use 10 times log₁₀ of a power ratio. Ten times the power is 10 dB. Doubling the power is 10 x log₁₀(2), which is 3.01 dB, which is why engineers talk about "3 dB" as the point where something halves. Earthquake magnitude works the same way: one whole step is ten times the ground motion.

How to use the log calculator

There are two inputs. The number you want the log of, which must be greater than zero, and the base, which must be positive and not equal to 1. Leave the base blank or set it to 10 for a common log, set it to e for a natural log, or type any other value for anything else. The result is the exponent.

Check your answer by reversing it. If the tool says log₃(81) is 4, confirm that 3 to the power 4 is 81. That habit catches base entry mistakes instantly, and it is worth keeping alongside the other everyday maths tools you use for homework checking.

Common questions

What is the difference between log and ln? Only the base. log with nothing written means base 10 in most contexts, ln always means base e. Some programming languages use log to mean natural log, which trips people up regularly.

Why do I get an error for log(0) or log of a negative number? Raising a positive base to any real power gives a positive answer, so no exponent can land on zero or on a negative number. As the input shrinks towards zero the log heads towards negative infinity, so there is no finite value to return. Negative inputs need complex numbers.

What is log of 1? Zero, in every base, because anything to the power zero is 1. Similarly log_b(b) is always 1.

What does antilog mean? It is the reverse operation: raising the base to your answer. The antilog of 2.699 in base 10 is 500, which just undoes the log you took.