aⁿ = x

log_ax = n

ELI10 TL;DR: They’re the opposite of exponents. If 10³ = 1000, then Log₁₀(1000) = 3.

Logarithms answer the question: “How many times do you have to multiply a base number (the number below the log) with itself to get the number within the brackets?”. Usually just saying log means log₁₀, since we use the ten numerals 0 to 9 in our everyday lives.

So log₁₀ (1000) equals 3 because you have to multiply 10 by 10 by 10 to get 1000, that’s three tens.

log₂ (256) equals 8 because when you start at 1 and double it 8 times, it becomes 256.

ln is the natural logarithm, equal to log_e or log_2.718…. It works the same as the rest, but it’s useful since e is used in some math equations representing some real world scenarios.

I did logarithms “by wrote” without an intuitive understanding of them until the day I realized that the logarithm base 10 is the number of digits a number is in base 10 (minus one).

• log₁₀10=1
• log₁₀100=2
• log₁₀1,000=3
• log₁₀10,000=4

(But whereas “number of digits minus one” is the same for, say, 1,000 as it is for 9,999, logarithm is “smoothed”. So the logarithm of a number between 1,000 and 10,000 will be some number between 3 and 4.)

Similarly, logarithm base 2 is the number of digits in base 2 minus one. Logarithm base 16 is the number of digits in base 16 minus one. Etc.

Natural logarithm (log base Euler’s constant) is a little trickier to think in terms of, but technically it is the number of digits in base e minus one. Numerical bases that have fractional parts are a sensical concept.

Thanks to logarithms, when you add numbers, you can also multiply them. This is how a slide rule works.

log rhythms?

Logarithms are a question. The base of the logarithm is the base of an exponent. The question is what exponent is required to get the number within the logarithm.

Log₁₀(5)=x means 10^x = 5