In what way are exponential and log functions related?
Our teacher said that they are inverses of each other but did not explain why.
Let f(x)=y=a^x
taking log(base a) of both sides we get
logy=x
f^-1(x)=logx
base other than a could also be taken.
So log was needed to find the inverse.
(Remember taking log creates some restrictions as log is a function.)
loga X=p (a is base)
then a^p=X
hope this helps
Suppose you have a function $\displaystyle y = 2x + 1$. The inverse of this function is:
$\displaystyle x = 2y + 1$
$\displaystyle x - 1 = 2y$
$\displaystyle y = \frac{1}{2}x - \frac{1}{2}$
Notice that we switched the x to y and the y to x. Why? You should know that a function has a domain and range. The inverse function also has a domain and range, but what are they? The domain of the inverse function is the range of the original function and the range of the inverse function is the domain of the original function. Now we covered what inverse functions are, let's consider the case:
$\displaystyle y = e^x$
Finding the inverse is easy, right? It's just:
$\displaystyle x = e^y$
But now this is an implicit equation. We need to find y explicitly, and that's where we use logarithms:
$\displaystyle y = \log_e{x}$
Logarithms base e is commonly called the natural logarithm, and it's notation is ln.
$\displaystyle y = \ln{x}$
Thus, $\displaystyle e^x$ and $\displaystyle \ln{x}$ are inverses of each other.