Consider an exponential function of the form y = ab^{x}. "b" is usually referred to as the base and x as the exponent. If we substitute values for x we can quickly calculate values for y. For example let y = 2^{x}. Y is equal to 2 to the power x. We could perform similar operations with other bases (3, 4, 10, etc...) The first two columns of the table given below show the values of y for various values of x. As you can see from the table increasingly negative values of x lead to smaller and smaller values of y. The curve approaches the x axis asymptotically. This means it gets closer and closer to the x axis but doesn't cross it. On the other hand as x becomes positive and increases the value of y also increases.

If we now interchange x and y the equation we are looking at becomes x = a^{y}. Unfortunately we have not yet covered the techniques for solving this equation for y. So, let's **plug-in values of y and calculate the resulting values of x.** The last two columns of the table show the value of x for various values of y. There are several things that you should note from the data shown. First when y is negative x is positive but decreasing in value. As a matter of fact no matter how negative y becomes x is always positive, although it does get smaller and smaller.

The figure shown to the right of the table depicts a portion of the graph of each function and the line y = x. The line y = x is included because the two equations are reflections of each other across this line. Thus the point (0,1) on the curve y =2^{x} is reflected as the point (1,0) on the curve x = 2^{x}. These curves are inverses of one-another. In mathematics you write the second equation as follows y= log_{2}x. In common language this says y is the log of x to the base 2. What it means is that y is equal to 2 to the power x (y = 2^{x}).

Value of x | Value of y | ------- | Value of y | Value of x |
---|---|---|---|---|

3 | 8 | 3 | 8 | |

2 | 4 | 2 | 4 | |

1 | 2 | 1 | 2 | |

0 | 1 | 0 | 1 | |

-1 | 1/2 | -1 | 1/2 | |

-2 | 1/4 | -2 | 1/4 | |

-3 | 1/8 | -3 | 1/8 |

If you look at your calculator you are likely to see keys labeled "log" and "ln". By common agreement log means base 10 and ln means base e (2.71.......). Logs to base 10 are called common logarithms and logs to base e (ln) are called natural logarithms. So y = log(x) means x = 10^{y} while y = ln(x) means x = e^{y}.
A few examples are probably in order at this point.

- common logarithm -- Log 100 = 2, or as a power --- 10
^{2}= 100- common logarithm -- Log 1 = 0, or as a power --- 10
^{0}= 1- common logarithm -- Log 0.1 = -1 or as a power --- 10
^{-1}= 1/10 or 0.1

Similarly

- natural logarithm -- ln 100 = 4.6052, or as a power -- e
^{4.6052}= 100- natural logarithm -- ln 1 = 0, or as a power -- e
^{0}= 1- natural logarithm -- ln 0.1 = -2.3026, or as a power -- e
^{-2.3026}= 0.1 (due to rounding 0.099999 on calculator)