Exponentiation and Logarithms with the L scale

prerequisites:

learn how to multiply and divide

learn how to find base 10 logarithms, and non-integer powers of 10

In the first case, notice that raising any base to a power is the same as raising 10 to a the power times a constant. In the second case, taking a log to any base is the same as taking a log base 10 and dividing by the same constant.

In both these cases, the constant is the log (base10) of the base you're switching to. So if you want to work with powers of 2, or logs base 2, you'd first figure out log(base10) 2 = 0.301, and keep that number handy.

To find y = a ^ x find log base 10 of a, ( described here ), call it k. multiply k and x, call it kx. y is 10 ^ kx ( described here )

To find y = log (base a) of x find log base 10 of a, ( described here), call it k find log base 10 of x, call it lx. y is lx divided by k

At first glance, these methods don't look much different from the sets of methods described in the more basic tutorials, but that's not the case. They are a pain. Learning to take logs and powers of 10 with the L scale takes practice - the decimal point rules need to be paid careful attention the first few times through. But worse than that, there's no way to 'chain' these operations. They generate intermediate results that are off the C/D scales, ( on L ). They need to be remembered or written down, then transferred to the C scale for multiplying/dividing. This reduces both precision and speed.

On the plus side, these methods are completely general. You can directly calculate 2 ^ 1,000,000 = 1.0 E 301030 just by applying the above methods. You can't read results like that off the LL scales.

Examples

9^5 to 3 decimal places

k = log10 of 9 = 0.9543, (direct lookup on L at 9C)

kx = 0.9543 x 5 = 4.771 (from C,D, setting decimal)

use L,C method for 10^kx

xi = 4, xm = 0.771

coefficient of answer is on C at 0.771L : 5.902

answer is 5.902E4, approx 59000

(exact answer is 59049)

0.09^-0.2

to find log10 of 0.09, we use the L,CI method:

0.09 = 9E-2

mantissa is on L at 9CI: 0.0458

k = log10 of 0.09 = -0.0458 + (-2) + 1 = -1.0458

kx = -1.0458 x -0.2 = 0.20916

10 ^ kx is a direct lookup on C at 0.20916L = 1.619

3.27^5.50

k = log10 of 3.27 is 0.514548 (direct lookup on L)

kx = 0.514548 * 5.5 = 2.83001 (from C,D and setting decimal)

use the L,C method for 10^kx:

xi = 2, xm = 0.83001

coefficient of answer is on C at 0.83001L : 6.76099

answer is 6.76099E2 = 676.099 approx 676.

The answer with no zooming is still 676, with some question about the final 6, and quite a bit faster.

The answer using a zoomed LL3_E scale is 676.10, much faster, but requiring an LL stack on your slide rule.

log(base 12) 16

to find k = log(base10) of 1.2E1, use the L,C method:

mantissa is at 1.2C on L: 0.0792

k = 0.0792 + 1 = 1.0792.

to find lx, use the L,C method again:

mantissa is at 1.6C on L: 0.2041

lx = 0.2041 + 1 = 1.2041.

final answer: lx / k = 1.2041/1.0792 = 1.116 (using division on C and D )

division refresher course images:

log( base 0.3 ) 0.4

to find k = log(base10) of 3E-1, use the L,CI method

mantissa is at 3CI on L: 0.5229

k = -0.5229 + (-1) + 1 = -0.5229

find lx = log(base10) 0.4 the same way: -0.3979

divide lx/k with C and D: answer 0.761