Logarithms - 3/21/13 - Jesse

This week we covered logarithms and logarithmic functions. A logarithmic function is the inverse of an exponential function.

Here’s an example of the exponential and logarithmic function relationship:  

y = 3 x   ---  Exponential Function

x = log3Y  ---  Logarithmic Function

In each function, the base is 3. These two functions are inverse of each other, meaning you can reflect the graph of y = 3 x  across the line y = x to obtain x =  log3Y .

If you wanted to graph the inverse relation of y = 3 x , first create a t-chart and plug in x value to get out y values. This will let you graph the exponential function. To graph the logarithmic (inverse) function, simply flip the t-chart’s x and y values. Here is what the two functions look like when graphed:

Sometimes you’ll see a logarithmic function that looks like this:

y = log X

When there is no number where the base is, then it is called the Common Log, and the base is assumed to be 10, which means:

y = log X    is really just  y = log10X

There is also a button on our graphing calculators that displays ln which is called the Natural Log. It represents log base e. So, for example,   ln X    =    logeX

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Properties of Logarithms

In class we went over the properties of logarithms. They can be found on page 331 in the book:

*Note that the Change-of-Base Formula can also be used for ln

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Homework Problems

The homework for the weekend was pg. 322   #51-55 odd, 65, 68   and   pg. 331   #1-23 odd.

Most of the homework required the use of logarithm properties to evaluate a function. I’ll do a few problems from the pages above.

From page 322:

55). Given   log20050     find the logarithm using the change-of-base formula

log20050 becomes (log50) / (log200)  --- do the division on your calculator and the answer is 0.74

From page 331:

3). Given   log5(5 * 125)    Use the Product Rule

log55 + log5125

=

1 + 3

= 4

5). Given  logt(8y)            Use the Product Rule

Logt(8) + Logt(y)

(leave it as that because it can’t be simplified any further)

13). Given  logt(m/8)       Use the Quotient Rule

logtm  -  logt8

19). Given  logb((p2q5) / (m4b9)     Use the Power Rule and the Quotient Rule

(2logbp + 5logbq)  -  (4logbm  -  9)