Question

what are logarithmic functions
please try to explain

Answer 1

What do you mean exactly by `what are' they?

Answer 2

It’s too hard to explain what logarithmic functions are in one sitting.
It took me a while to understand them.
And you can’t fully appreciate them until you learn what they really are at a deep level.

But a logarithmic function,
you can think of it as the inverse of the exponential function.

For example, consider f(x) = e^x.
That’s the exponential function.

You can plug in values for x, and you get back values f(x).
You can plug in x = 0, and you will get back f(0) = e^0 = 1.
So (0, 1) is a point on f(x) = e^x.
Plug some more in.
Plug x = 1, and you get back f(1) = e^1 = e. (1, e) is a point on f(x).
Plug x = 100, and you get f(100) = e^100. (100, e^100) is another point.
Plug in x = -100, and you get back e^(-100). (-100, e^(-100)) is another point.
If you try plug in all possible numbers into x for f(x) = e^x,
you’ll see that you trace out a smooth, continuous curve that “grows exponentially.”

The logarithmic functions are inverses of exponential functions.
Let me call g(x) to be g(x) = log x, where the log has “base e,” the natural logarithm.

Remember that (1, e) is a point on f(x)?
Then (e, 1) is a point on g(x).
That’s because g(e) = log (e) = 1.

Similarly, since (100, e^100) is a point on f(x),
we have (e^100, 100) is a point on g(x).
That’s because g(e^100) = log (e^100) = 100 * log (e) = 100.

Also, since (-100, e^(-100)) is a point on f(x),
we have (e^(-100), -100) is a point on g(x).
That’s because g(e^(-100)) = log (e^(-100)) = -100 * log (e) = -100.

So if a point (a, b) is on f(x) = e^x,
then the point (b, a) is on g(x) = log (x).
So we see that they’re mirror images of each other.
In fact, they’re mirror images across the line y = x since the
x and y coordinates swapped.

So e^x and log x are inverse functions not only because
they “undo” each other, but also because
they’re mirror images of each other across the line y = x.

This helps us visualize log x,

Graphically, inverse functions are mirror images of each other,
reflected across the line y = x.
So imagine drawing f(x) = e^x on a graph paper.
Then draw in the line y = x.
Fold your graph paper along the line y = x, and let
f(x) = e^x be reflected across y = x.
The reflection of e^x across the line y = x is just
where the the curve e^x touches the graph paper when you folded it.

That reflection is g(x) = log x.