I understand the following two identities, however could use guidance on on this applies to multiplication of numbers. I think the antilog() gets applied somewhere after the result of $log_b(x) + log_b(y)$ is established, but am not sure.
Can anyone provide an example?
Thank you
On
So a slide rule has two slats with markings. From $1$ to $100$ (two inches apart; $10$ is one inch after $1$ and $100$ is one inch after $10$) but not evenly spaced and
If we want to multiply $8792$ by $14593850$ we look up the marking of $8.79$. It happens to be $0.94$ inches on the ruler. (Because $\log 879\approx 0.94$). And we slide the other slat's $1$ position to line up to the $8.79$ marking.
On the second slat we look up the $1.46$ marking. It happens to be $0.16$ inches along the slat. We see were that lines up with the first slat. It is $0.94 + 0.16 = 1.10$ inches along. And it is marked as $12.59$. (Because $10^{1.1}\approx 12.59$; that's how these markers were marked).
So we figure $8.792\times 10^3 \cdot 1.4593850\times 10^7= 12.59 \times 10^{10}$.
And indeed $8792\times 14593850=128309129200$
Here's a rather simplistic example: suppose that you want to multiply 1000 by 100000. Let's use base-10 logs :-) The log of 1000 is 3, the log of 100000 is 5. Adding these two we get 8. Then we take the antilog of 8: that's $10^8$ - lo and behold, that's the product of 1000 and 100000.
That's the exact same procedure for other numbers, but for those you'll have to use the log tables to figure the log of the number and then after adding them together, use the log tables "backwards" to calculate the antilog. Of course, you can use your calculator instead of log tables, although you cannot use your calculator to multiply the two original numbers together: that would be cheating :-)