log17(y) = log17(17^x)
Log base 17 will tell us the power that we have to raise 17 to, to get some number. In this case, that number is 17^x. Clearly, the power that we must raise 17 to to get 17^x is x. Therefore
log17(y) = x
My calculator doesn't have a log17 button, so I'd like to convert that into log10, which thankfully is as easy as rewriting it like so
log10(y) / log10(17) = x
or in shorthand
log(y) / log(17) = x
So if I ever need to raise 17 to a power and hit some target number y, that's how I do it. You can see it works by plugging in, say, 1400 for y:
log(1400) / log(17) = 2.55689446...
17^2.55689446... = 1400
When I ask my question, "To what exponent must I raise this base to get some number?" the log function tells me an answer which is an exponent. Because logs always spit out exponents, all the cool tricks we have for working with exponents are true for logs as well. For example
10^a * 10^b = 10^(a+b)
just like
log(a) + log(b) = log(a*b)
and some other cool things.
edit-
log(base 10)(3x+5) = 8
to solve this, we go the other way. log10(3x+5) asks, 10^? = 3x+5
Without knowing what that question mark is, if we raise 10 to that power, we get 3x+5. So let's simplify by raising 10 to both sides of the equation-
10^(log10(3x+5)) = 10^8
and rewrite to simplify
3x+5 = 10^8 = 100000000
then we can solve by algebra. x works out to be about 33333331.66666667
You can check your work by plugging that back into the original equation
log( ( 3 * 33333331.7 ) + 5) = 8