Exponential and logarithmic equations
THE LOGARITHMIC FUNCTION WITH BASE b is the function y = logb x. b is normally a number greater than 1 (although it need only be greater than 0 and not equal to 1). The function is defined for all x > 0. Here is its graph for any base b.
Note the following:
• For any base, the x-intercept is 1. Why?
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The logarithm of 1 is 0. y = logb1 = 0.
• The graph passes through the point (b, 1). Why?
The logarithm of the base is 1. logbb = 1. • | The graph is below the x-axis -- the logarithm is negative -- for | | | 0 < x < 1. | | | Which numbers are those that have negative logarithms? |
Proper fractions. • | The function is defined only for positive values of x. | | | logb(−4), for example, makes no sense. Since b is always positive, no power of b can produce a negative number. |
• The range of the function is all real numbers.
• The negative y-axis is a vertical asymptote (Topic 18).
Example 1. Translation of axes. Here is the graph of the natural logarithm, y = ln x (Topic 20).
And here is the graph of y = ln (x − 2) -- which is its translation 2 units to the right.
The x-intercept has moved from 1 to 3. And the vertical asymptote has moved from 0 to 2.
Problem 1. Sketch the graph of y = ln (x + 3).
This is a translation 3 units to the left. The x-intercept has moved from 1 to −2. And the vertical asymptote has moved from 0 to −3.
The exponential function with positive base b > 1 is the function y = bx.
It is defined for every real number x. Here is its graph:
There are two important things to note:
• The y-intercept is at (0, 1). For, b0 = 1.
• The negative x-axis is a horizontal asymptote.