How to find values such as 6 to the power 0.7
Calculating 10 ^ 0.1 , 0.2, 0.3, etc We talked about mentally calculating logarithms. Now, let's go the other way.
How does one mentally calculate 10^0.1, 10^0.2, etc.?
You could of course memorize the numbers, but this is the mental calculation subforum, so we need to calculate them. Mentally. Ouch.
Think about it; how would you calculate 10^0.1?
It is - by definition - the tenth root of 10, so one option is to try a number and raise that to the tenth power - for example by repeatedly squaring. And then iteratively refine the initial guess.
This takes time. It is great training of course, but let's find an easier way.
We know that log(2) = 0.301. 0.30103 if you want to be precise.
Since this number is very close to 0.3, we could use this number and make a small correction for the fact that we are working with 0.301 instead of 0.3.
If we do this we can calculate these numbers with about 4 digits precision.
As always, it is a lot more work to explain than to do, so bear with me.
If log(2) = 0.301, this means that 10^0.3 is almost 2.
If we do the correction later, we can start by stating that 10^0.3 = 2.
Then, 10^0.6 = 4 and 10^0.9 = 8.
Now, from the nine numbers 0.1 - 0.9, we have already 3 numbers covered.
Let's continue adding 0.3:
10^1.2 = 16,
10^1.5 = 32,
10^1.8 = 64,
10^2.1 = 128,
10^2.4 = 256,
10^2.7 = 512.
That was not difficult, right?
Let's bring the exponent under 1:
If 10^1.2 = 16, then 10^0.2 = 1.6
If 10^1.5 = 32, then 10^0.5 = 3.2
If 10^1.8 = 64, then 10^0.8 = 6.4
If 10^2.1 = 128, then 10^0.1 = 1.28
If 10^2.4 = 256, then 10^0.4 = 2.56
If 10^2.7 = 512, then 10^0.7 = 5.12
You probably see where this is going.
To finish the series we can add:
10^0.0 = 1
10^1.0 = 10
Putting them all in order we get:
10^0.0 = 1
10^0.1 = 1.28
10^0.2 = 1.6
10^0.3 = 2
10^0.4 = 2.56
10^0.5 = 3.2
10^0.6 = 4
10^0.7 = 5.12
10^0.8 = 6.4
10^0.9 = 8
10^1.0 = 10
Take a moment to see what has happened here. We calculated all these numbers by adding 0.3 on one side and doubling the last number on the other side.
This can be done in a matter of seconds.
If you want to know 10^0.4, all you need to know is 'what multiple of 3 ends in a 4?'
Well; 3 x 8=24.
So, we take .3 x 8 = 2.4:
10 ^( .3 * 8 ) = 10 ^2.4 = 2 ^8 = 256 => 10.4 = 2.56
That version is the long version. The short version goes like this:
What multiple of 3 ends in 4? Well, 24 (8 times 3) => use 8.
2^8 = 256, so 2.56.
Done.
Try this out for yourself with a couple of numbers .
We talked about mentally calculating logarithms. Now, let's go the other way.
How does one mentally calculate 10^0.1, 10^0.2, etc.?
You could of course memorize the numbers, but this is the mental calculation subforum, so we need to calculate them. Mentally. Ouch.
Think about it; how would you calculate 10^0.1?
It is - by definition - the tenth root of 10, so one option is to try a number and raise that to the tenth power - for example by repeatedly squaring. And then iteratively refine the initial guess.
This takes time. It is great training of course, but let's find an easier way.
We know that log(2) = 0.301. 0.30103 if you want to be precise.
Since this number is very close to 0.3, we could use this number and make a small correction for the fact that we are working with 0.301 instead of 0.3.
If we do this we can calculate these numbers with about 4 digits precision.
As always, it is a lot more work to explain than to do, so bear with me.
If log(2) = 0.301, this means that 10^0.3 is almost 2.
If we do the correction later, we can start by stating that 10^0.3 = 2.
Then, 10^0.6 = 4 and 10^0.9 = 8.
Now, from the nine numbers 0.1 - 0.9, we have already 3 numbers covered.
Let's continue adding 0.3:
10^1.2 = 16,
10^1.5 = 32,
10^1.8 = 64,
10^2.1 = 128,
10^2.4 = 256,
10^2.7 = 512.
That was not difficult, right?
Let's bring the exponent under 1:
If 10^1.2 = 16, then 10^0.2 = 1.6
If 10^1.5 = 32, then 10^0.5 = 3.2
If 10^1.8 = 64, then 10^0.8 = 6.4
If 10^2.1 = 128, then 10^0.1 = 1.28
If 10^2.4 = 256, then 10^0.4 = 2.56
If 10^2.7 = 512, then 10^0.7 = 5.12
You probably see where this is going.
To finish the series we can add:
10^0.0 = 1
10^1.0 = 10
Putting them all in order we get:
10^0.0 = 1
10^0.1 = 1.28
10^0.2 = 1.6
10^0.3 = 2
10^0.4 = 2.56
10^0.5 = 3.2
10^0.6 = 4
10^0.7 = 5.12
10^0.8 = 6.4
10^0.9 = 8
10^1.0 = 10
Take a moment to see what has happened here. We calculated all these numbers by adding 0.3 on one side and doubling the last number on the other side.
This can be done in a matter of seconds.
If you want to know 10^0.4, all you need to know is 'what multiple of 3 ends in a 4?'
Well; 3 x 8=24.
So, we take .3 x 8 = 2.4:
10 ^( .3 * 8 ) = 10 ^2.4 = 2 ^8 = 256 => 10.4 = 2.56
That version is the long version. The short version goes like this:
What multiple of 3 ends in 4? Well, 24 (8 times 3) => use 8.
2^8 = 256, so 2.56.
Done.
Try this out for yourself with a couple of numbers .
