Locate the common log key on your calculator. Using it, find the common logarithms of the number listed in table 1.

Logarithms Data

 

 

Procedure:

 

Part A.  Common and Natural Logarithms and Their Antilogarithms

 

1. Locate the common log key on your calculator. Using it, find the common logarithms of the number listed in table 1.

 

2. Now locate the natural log key and find the natural logarithms of the values in table 2.

 

 

Table 1			Table 2
X	Log(X)		X	ln(X)
345.1			534.2
2.3 x 108			9.11 x 10-28
0.657			1000
1.67 x 10-27			3.7 x 109
2.33			1.12

 

 

3. Antilogarithms in the common log system are obtained by raising 10 to corresponding logarithmic values. Remember, an antilogarithm is the number that the logarithm represents. Locate the 10^x key on your calculator.  Most models will show display it as an upper case function to the log key.  Using this function, find the antilogarithms of the logarithms listed in table 3.

 

Table 3			Table 4
Log(X)	X		Ln(X)	X
22.1			3.22
-2.30			-34.21
3.00			6.91
-8.62			-60.3
0.114			0.370

 

4. Antilogarithms in the natural log system are determined with the function e^x. Locate this key on your calculator.  Most models will display it as an upper case function to the ln(x) key.  Use this function to find the antilogarithms of the logarithms listed in table 4

 

 

Part B.  Algebraic Applications

 

1. The activity version of the decay law is given by

 

 

Using the common log identities, show that

 

or

since .

 

Work Space:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2. Show that in the natural log system.  Remember.

 

Work Space:

 

 

 

 

 

 

 

3. The inverse square law of intensity is given by

 

Show that

 

 

Work Space:

 

 

 

 

 

 

 

 

 

 

 

 

 

Part C.  Graphical Applications

 

1. Consider the relationship. Find the values of y for the values of x listed in table 5.  Using rectangular graph paper, make a graph of the data listed in table 5.

Table 5			Table 6
X	Y		X	Log(Y)
1			1
2			2
3			3
4			4

 

2. Now we want to construct a log graph for. Since the independent variable is already an exponent, you only need to find the logarithm of the y-values.  Fill in table 6 using the y-values in table 5.  Make a plot of table 6 on rectangular graph paper.  The graph is semilogarithmic, and the points should conform to a straight line.

 

3. Consider the relationship. Find the values of y for the values of x listed in table 7.  Using rectangular graph paper, make a graph of the data in table 7.

Table 7			Table 8
X	Y		Log(X)	Log(Y)
2
4
6
8

 

4. Neither variable is an exponent in. Therefore, to make a log graph, the logarithms of both variables must be plotted.  Fill in table 8 using the x and y values in table 7.  Make a plot of the entries in table 8 on rectangular graph paper.  The points should fall along a straight line on this log-log graph.