Mental Calculation of Logarithm and Antilogarithm explained step-by-step

(1) LOGARITHM

We search for Logarithm of 87:

We must search for a logarithm close to logarithm of 87.

Log of 87 = Log of 88 - y

This y we will found it doing 88/87

88/87 = 1.0114

1.0114 is the 1.14%

So, logarithm of 87 = Logarithm of 88 - 1.14%

But 1.14% of what? The answer is 1.14% of the amount, and every 1% is 0.00432 times. No need to understand why, only remember 0.00432. So, 0.00432 x 1.14 = 0.00492

We factorize 88 as 11 x 8 and by properties of logarithms we know that a multiplication turn into addition. So logarithm of 88 = Logarithm of 11 + Logarithm of 8.

Logarithm of 11 + Logarithm of 8 = 1.94448

LOGARITHM OF 87 = LOGARITHM OF 88 - 1.14% OF 0.00432 = 1.94448 - 0.00492 = 1.93956

(2) ANTILOGARITHM

We search for 10^0.87

We must search for the logarithm close to the exponent. We know that the logarithm of 7 is in the range of this exponent; in fact, Logarithm of 7.4 is close, because is 0.86923

We subtract the exponent minus logarithm close to the exponent: 0.87 - 0.86923 = 0.00077

Now, we must factorize 0.87 as 10^0.86923 x 10^0.00077

We already know that 10^0.86923 is 7.4, so we can factorize 0.87 as 7.4 x 10^0.00077

We must do the division 0.00077/0.00432, because we want to know how much 1% is included in 0.00077. The result is 0.178

What is 0.178% of 7.4? It is 0.00178 multiplied by 7.4, that is 0.00178 x 7.4 = 0.01317

Lastly, 0.01317 added to 7.4, that is 0.01317 + 7.4 = 7.41317


@Daniel_360 and @Kinma, based in Daniel’s work, we have calculated both Logarithm and Antilogarithm.

Of course I was helped by someone, that’s not made by my intelligence.

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It’s a great post!
I have nothing to add.

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Thanks!

Or maybe I do…

This is the same way I work.
If I need a logarithm of a number that I do not know, then I start with a number close to it and work out the difference (in percentages).

And indeed, if the difference is a small percentage, you can just assume the graph to be linear.
And work out the percentage times 0.00432.

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