I'm learning about this in class. The teacher switches ln for e^x and the numbers all switch. I'm just confused as to why this works.

Ben,

Here's another perspective:

You know how to graph y = b^x, a simple exponential function.

You also know how to find the inverse of a function by reversing the x and y (this reflects the graph of the function over the line y = x).

If you try to find the inverse of the simple exponential function you get x = b^y. It's easy to graph this function: you just take all the points you used to graph y = b^x and swap the coordinates, graph those and draw a smooth curve through them. If you
do it you'll see the graph of the inverse is a reflection over y=x of the exponential function.

But the problem is, how do you solve for y in the inverse function x = b^y? There is no way you can do it without defining a new function name; that new function name is the "logarithm". The logarithm function has the same information in it as does x =
b^y, i.e., y = log_b(x). See the x, y, and b in both expressions? (The b is printed as a subscript usually, I'm using "_b" to indicate subscript.)

You may be given one of these expressions and asked for the equivalent other one; just identify the three numbers in the given expression and put them in the right place in the other.

There are two shorthand notations when the base is 10 or e: log and ln. You will see these on calculators.

Finally, please always use parenthesis to enclose the input to any function, including logs. E.g., log_3(27), not log_327.

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