How we can show a^loga b =b , e^lnb=b thanks a lot.?

Question

How we can show a^loga b =b , e^lnb=b thanks a lot.?

Answer 1

For the first equation, take the loga of both sides of the equation. Since loga(a^x) = x is the definition of a logarithm, you get this:

loga (a^loga b) = loga(b)

loga b = loga b

For the second one, do exactly the same thing, except take the log base e (ln) of both sides:

ln (e^ln b) = ln(b)

ln b = ln b

Alternatively, you can immediately see that both statements are true simply by remembering that the exponential function and the logarithm function are inverses of each other. In both of these examples you take a number, take the logarithm of it, and then take the exponential of the result. This will always give you back the same number you started with. It's exactly like taking a number, doubling it, and then taking half of the result. You will end up right where you started.