find the logbase7 12?

Modern calculators let you take logs to any base you like. If you're working with an older type of calculator, or (say) the Windows calculator, you're confined to using log to base 10 or base e.

Suppose you have to work in base 10. Then use what I call the "ABBA" log conversion formula:

log[a]x = log[b]x / log[b]a.

In this case, put a = 7, b = 10 and x = 12, so that

log[7]12 = log[10]12 / log[10]7, and you can work out both top and bottom of this fraction on the calculator.

in other words, find x:

7^x = 12

assuming you don't have the base 7 log function on your calculator, let's proceed with common log (i.e. base 10 log):

first take log of both sides:

log(7^x) = log(12)

using known properties of logs (which you should be able to prove are true), we can write the equation like this:

x*log(7) = log(12)

divide both sides by log(7)

x = log(12) / log(7) = approximately...wait, what do we expect it to be?

7^1 = 7 and 7^2 = 49, so since 12 is between 7 and 49, we expect the result to be between 1 and 2 and probably closer to 1.

using calculator we get about 1.277