In the adjoining diagram there are two curvation graphs of logax and logb x shown for x>0. If (a,b) > 0 and a=! b =!1, then:
Answers
Answer:
Topic 20
• 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.
Lesson 20 of Arithmetic
• 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).
Graph of logarithm
And here is the graph of y = ln (x − 2) -- which is its translation 2 units to the right.
Graph of logarithm
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).
Graph of logarithm
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.
Exponential functions
By definition:
logby = x means bx = y.
Corresponding to every logarithm function with base b, we see that there is an exponential function with base b:
y = bx.
An exponential function is the inverse of a logarithm function. We will go into that more below.
An exponential function is defined for every real number x. Here is its graph for any base b:
Graph of exponential
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. For, when x is a large negative number -- e.g. b−10,000 -- then y is a very small positive number.
Problem 2.
a) Let f(x) = ex. Write the function f(−x).