How can I calculate a logarithm without a calculator?

Since you were not specific as to what base of logarithm you wanted, I will take the liberty of showing you how to calculate logarithms base 2 in binary. They are perhaps the easiest to do by hand. Elsewhere I have shown a method to calculate common (base `10`) logarithms, but that involves a lot of raising numbers to the `10`th power so is rather tedious.

Let us calculate `log_2 7`.

From now on express numbers in binary...

We want to calculate `log_(10_2) 111_2`

`10_2^(10_2) = 100_2 < 111_2 < 1000_2 = 10_2^(11_2)`

So the digits before the binary point are `color(blue)(10)`

Next divide `111_2` by `10_2^(10_2)=100_2` to get `1.11_2`

Square `1.11_2` to get `11.0001_2`

Since this is greater than `10_2`, the first digit after the binary point is `color(blue)(1)`

Divide by `10_2` to get `1.10001_2`

Square `1.10001_2` to get `10.0101100001_2`

Since this is greater than `10_2`, the second digit after the binary point is `color(blue)(1)`

Divide by `10_2` to get `1.00101100001_2`

Square `1.00101100001_2` to get `1.0101111111011011000001_2`

Since this is less than `10_2`, the third digit after the binary point is `color(blue)(0)`

Square `1.0101111111011011000001_2` to get `1.11100011100110100101000001010111110110000001_2`

Since this is less than `10_2`, the fourth digit after the binary point is `color(blue)(0)`

To cut down on the arithmetic, I will approximate this as `1.11100011100110100101_2`

Square `1.11100011100110100101_2` to get `11.1001000110001111101001101110010001011001_2`

Since this is greater than `10_2`, the fifth digit after the binary point is `color(blue)(1)`

Divide by `10_2` to get `1.11001000110001111101001101110010001011001_2`

I'll stop here, but I hope you get the idea.

Putting the digits we have found together we get:

`log_(10_2) 111_2 ~~ 10.11001_2 = 2+25/32 ~~ 2.8`

Actually `log_2(7) ~~ 2.80735`