The logarithmic function as the inverse of the exponential function
The graph of the exponential function y = 2x lets you determine logarithms. For example,
| if y = 4, then x = 2 | that is, log24 = 2 | if y=2, then x = 1 | log 22 = 1 |
, You obtain in this way the logarithms corresponding to the base 2. If you want logarithms to another base, for example 10, you must use the graph of y=10x. In other words, you can use every exponential function y=gx, in order to find graphically the logarithms of a number to the base g. The independent variable of the exponential function becomes the dependent variable. The logarithmic function is y = loggx or gy = x, the corresponding exponential function is gx=y.You see how a new function arises by interchange of the dependent and independent variables y and x; you obtain thus the inverse function and the Rule:
The
image of the logarithmic function is the reflection
of the corresponding exponential function in the straight line y
= x.
Naturally, the graph of the logarithmic function allows you to extract the logarithms of
all numbers just like out of the exponential function. In fact,
for a base g (g > 0, 
1):
Every
positive number has a logarithm;
zero and negative numbers do not have logarithms.
You always have logg1 = 0. The logarithm increases with the argument x. For 0<x< 1, the logarithms are negative. The section of the curve, which lies in the fourth quadrant, approaches in the downward direction the ordinate axis without ever reaching it. The section in the first quadrant goes away from the abscissa with growing x and increases ever more slowly.