### The Relationship Between Exponential and Logarithmic Functions

Consider an exponential function of the form y = abx. "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 = 2x. 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 = ay. 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 =2x is reflected as the point (1,0) on the curve x = 2x. These curves are inverses of one-another. In mathematics you write the second equation as follows y= log2x. 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 = 2x).

Value of xValue of y-------Value of yValue of x
3838
2424
1212
0101
-11/2-11/2
-21/4-21/4
-31/8-31/8

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 = 10y while y = ln(x) means x = ey. A few examples are probably in order at this point.

• common logarithm -- Log 100 = 2, or as a power --- 102 = 100
• common logarithm -- Log 1 = 0, or as a power --- 100 = 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 -- e4.6052 = 100
• natural logarithm -- ln 1 = 0, or as a power -- e0 = 1
• natural logarithm -- ln 0.1 = -2.3026, or as a power -- e-2.3026 = 0.1 (due to rounding 0.099999 on calculator)