# Calculate log43 and log85 mentally

*Log43

3 + 40 ==> 3/40% ==> (3/0.4) × 0.00432 ==> 0.0324 + 1.6020 = 1.6344

Calculator give me 1.6334

*Log85

5 + 80 ==> 5/80% ==> (5/0.8) × 0.00432 ==> 0.0270 + 1.9030 = 1.9300

Calculator gives me 1.9294

How to calculate log 43?

First option is to take log 40. However; 43 is 7.5% higher, so we need to to add log 1.075 \approx 3 / 4 * log(1.1) = 3/4 * 0.0414.
I would do the division by 4 first: 0.0414/4 = 0.01035.
Then multiply by 3.
0.01035 * 3 = 0.03105
log 40 = 1.60206, since log 40 = log 10 + log 2 + log 2

Add the two together to get:
1.60206 + 0.03105 = 1.63311

10^{1.63311} = 42.965, so pretty close to 43!

A more accurate method is this:
43^3 = 79507
This number - 79507 - is almost 80,000 - 500.
This difference of 500 is 0.62% of 80,000.
Any difference smaller than 1% will lead to very accurate logarithms.

So we take log 80,000 (= 4.90309, because 80,000 = 10,000 * 2^3), and subtract 0.62% = 0.62 * log 1.01.
0.62 * 0.00432 = 0.0026784
4.90309 - 0.0026784 = 4.9004116

If log(43^3) = 4.9004116 then
log(43) = 4.9004116 / 3
4.9004116 / 3 = 1.63347053

The calculator gives 10^{1.63347053} = 43.0002, so the log is very accurate.

85 is 17 * 5, so if you remember both log17 and log5, which we calculated here:

then you can just add the two numbers together.

Another way to calculate log 43 is to add log 42 and log 44 together.
since 44 = 4 * 11 and 42 = 6 * 7 these should be easy to add if you know the smaller logs.

That is the same number as log(43^2 - 1).

43^2 = 1849 and 42*44 = 1848, so you need to make a correction of 1/1848, or 0.054% = 0.54 * 0.000434.

Now you know log(1849) = log(43^2) = 2 * log 43

So divide the number by 2 and you know log 43.