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# Logarithms and logarithmic functions

## on Jul 16, 2010

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## Logarithms and logarithmic functionsPresentation Transcript

• Introduction To Logarithms
• Why is logarithmic scale used to measure sound?
• Our first question then must be: What is a logarithm ?
• One one Function
• (3,8) (2,4) (1,2) (-1,1/2) (-2,1/4) (8,3) (4,2) (2,1) (1/2,-1) (-1/4,-2) Inverse of
• The inverse of is the function
• (3,8) (2,4) (1,2) (-1,1/2) (-2,1/4)
• Domain of logarithmic function = Range of exponential function = Range of logarithmic function = Domain of exponential function =
• 1. The x -intercept of the graph is 1. There is no y -intercept. 2. The y -axis is a vertical asymptote of the graph. 3. A logarithmic function is decreasing if 0 < a < 1 and increasing if a > 1. 4. The graph contains the points (1,0) and (a,1). Properties of the Graph of a Logarithmic Function
• Logarithmic Abbreviations
• log 10 x = log x (Common log)
• log e x = ln x (Natural log)
• e = 2.71828...
• Of course logarithms have a precise mathematical definition just like all terms in mathematics. So let’s start with that.
• Definition of Logarithm Suppose b>0 and b≠1, there is a number ‘p’ such that:
•
• The first, and perhaps the most important step, in understanding logarithms is to realize that they always relate back to exponential equations.
• You must be able to convert an exponential equation into logarithmic form and vice versa. So let’s get a lot of practice with this !
• Example 1: Solution: We read this as: ”the log base 2 of 8 is equal to 3”.
• Example 1a: Solution: Read as: “the log base 4 of 16 is equal to 2”.
• Example 1b: Solution:
• Okay, so now it’s time for you to try some on your own.
• Solution:
• Solution:
• Solution:
• It is also very important to be able to start with a logarithmic expression and change this into exponential form. This is simply the reverse of what we just did.
• Example 1: Solution:
• Example 2: Solution:
• Okay, now you try these next three.
• Solution:
• Solution:
• Solution:
• We now know that a logarithm is perhaps best understood as being closely related to an exponential equation. In fact, whenever we get stuck in the problems that follow we will return to this one simple insight. We might even state a simple rule.
• When working with logarithms, if ever you get “stuck”, try rewriting the problem in exponential form. Conversely, when working with exponential expressions, if ever you get “stuck”, try rewriting the problem in logarithmic form.
• Let’s see if this simple rule can help us solve some of the following problems.
• Solution: Let’s rewrite the problem in exponential form. We’re finished !
• Solution: Rewrite the problem in exponential form.
• Example 3 Try setting this up like this: Solution: Now rewrite in exponential form.
• These next two problems tend to be some of the trickiest to evaluate. Actually, they are merely identities and the use of our simple rule will show this.
• Example 4 Solution: Now take it out of the logarithmic form and write it in exponential form. First, we write the problem with a variable.
• Example 5 Solution: First, we write the problem with a variable. Now take it out of the exponential form and write it in logarithmic form.