How do you convert base ten numbers to base fifteen?

Counting in base-15 would be something like:

1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, 10,

11, 12, 13, 14, 15, 16, 17, 18, 19, 1A, 1B, 1C, 1D, 1E, 20

... and so on.

By tradition, letters starting with "A" are used for digits representing numbers beyond 9.

26 in base 10, converted to base 15, would have one in the 15s place, and the digit representing 11 in the ones place: 26[10] = 1B[15]

How to convert is that you break up the number into powers of 15, just like a base-10 number is broken up into digits representing multiples of powers of 10.

If your number is over 225 (15^2), you'd have to compute the digit for the 225s place. Then you'd take the remainder and divide by 15, to compute the digit for the 15s place, and the remainder from that calculation would be the digit for the ones place.

First you need five more digits. Let's call them A, B, C, D, and E, so that A = 10, B = 11, C = 12, D= 13, and E = 14.

Then for any number x in decimal, if you divide x by 15, the remainder will give you the last digit and the quotient (converted to base 15) will give you the rest of the digits.

Let's try 26 base 10. 26 = 1* 15 + 11. So in base 15, this is 1B.

Now let's try 322 base 10. 322 = 21*15 + 7. So 7 is the last digit in base 15, and to find the rest of the number we convert the quotient, 21 to base 15. 21 = 1*15 + 6. So 322 base 10 is equal to 167 base 15.

In base 10: the number 26 means 2*10 + 6*1

In base 15: the same number would be represented by 1*15 + 11*1. But we don't have a single digit for 11, so we typically use the letter B. (A = 10, B = 11, etc).

So the base ten number 26 would be 1B in base 15.

When 26(base 10) is converted into base 15. The ones place equals to 15 to the zeroth power, or 1, the second place is 15 to the first power or 15. Let's think:

26=15 times 1 plus 11. 26(base ten)=1B(base fifteen).

When 26 (base 15) is converted into base 10. 2 times 15 is 30, plus 6 times 1 equals to 36 (base 10).

Let y = log to the base a of x

Then a^y = x

Which means the power of a (which is the base ) is the value of x to the base a.

Note :

Let a = 15, and x = 26

Then 15^y = 26

To find y :

Take the log to the base 10 on both sides :

log 15^y = log 26

y log 15 = log 26

y = log 26 / log 15

So the value of y is the value of 26 to the base 15.

If you are confused : Take a simpler example...

We know that :

10^2 = 100 where the base is 10

Clearly, log 100 = power of 10 on the LHS which is 2.

it is 20