Let’s first observe this example with a base of 10.

3.4 Evaluating Logarithms and Graphing Logarithmic Functions

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Logarithms are important in many applications of mathematics to everyday problems, particularly in biology, engineering, economics and social science. In this section we will show how to find a numerical approximation for a logarithm. Because the number system we use is a base 10 system, we will then use logarithms with a base of 10. Logarithms to base 10 are called common logarithms and log10x is abbreviated as simply log x, where the base is understood to be 10.

Let’s first observe this example with a base of 10.

10Solve: log 100 x

We can solve this by changing from logarithmic form to exponential form to solve.

x10 =100

x 210 =10

x=2 However, this problem could have been solved by another method. Notice in this example, the base is 10. Also, our calculators work with a base of 10. Therefore, we can use our calculator to solve this equation.

Next Slide

To evaluate log10100, find the log button on your calculator. Press the log button then 100 and the equal or enter button. The answer will be 2. On some calculators, you may need to type in the 100 and then press the log button.


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log1.98 0.2967

Example 1. Evaluate log 1.98 rounded to four decimal places.

Answer:

Round to 0.2967 because the directions are to round to 4 decimal places. We could perform a check to be confident of our answer. Remember the base is understood to be 10.

0.2967Evaluate 10 on your calculator.

The result is not exactly 1.98 because we did round off to 4 decimal places.

Use a calculator to find each common logarithm. Express answer to four decimal places.

Your Turn Problem #1

a) log67.5

b) log0.415Answer: a) 1.8293

b) 0.3820


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0.295610 =x

There are two procedures for evaluating 100.2956 on your calculator. Type in 10, press the exponent button, then type in 0.2956. The second method is to use the 10x key (2nd then log) on your calculator.

Solve for x: logx 0.2956 (Round to five significant deciExampl mal ple 2. aces)

If the base is not written, then it is assumed to be 10.

10log x 0.2956

x 1.97514 962206

Therefore x=1.9751 rounded to five significant digits.

Note: significant digits includes digits in front of the decimal point.


Use your calculator to find x when given log x. Round to five significant digits.a) logx 1.4023

b) logx 2.3751

Answer:

a) 25.252b) 0.0042160

Note: The two zeros before the four are not significant digits.


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If an expression is written such as ln 241.7, the base is assumed to be e. This expression can be found by using the LN key on your calculator. You should get 5.4877 (rounded to four decimal places) This means e to the power 5.4877 is approximately 241.7.

eln 241.7 xxe 241.7

Since x = 5.4877, e5.4877241.7.

Algebraically, we can verify.

The natural logarithm is a logarithm with a base of e. ‘e’ is approximately 2.718. You can get this number on your calculator by pressing the ex on your calculator and then using an exponent of 1 to obtain a numerical approximation. You may need to press shift or 2nd first if the ex key is in small yellow writing above the LN key. The LN key stands for natural logarithm.

The Natural Logarithm

Example 3. Evaluate ln 225 rounded to four decimal places.

Answer:

We could perform a check to be confident of our answer. Remember the base is understood to be e. (Instead of using e, you could also use 2.718 to get an estimate.)

5l .n2 125 416

5.4161Evaluate e on your calculator.

Use a calculator to find each natural logarithm. Express answer to four decimal places.


a) ln67.5b) ln0.0415

Answers: a) 4.2121b) 3.1821


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3.4e =x

To evaluate e3.4 on your calculator, use the ex key on your calculator.

Solve for x: lnx 3.4 (Round to five significant deciExampl mal ple 4. aces)

The base for ln is assumed to be e. eln x 3.4

x 29.9641000474

Therefore x=29.964 rounded to five significant digits.


Use your calculator to find x when given ln x. Round to five significant digits.a) lnx 4.14

b) lnx 3.12

Answer: a) 62.803

b) 0.044157


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Graphing Logarithmic Functions

Let’s first graph the function f(x) = log3x. At this point, we can not use our calculators because the base is 3. However, we can change the notation from f(x) to y.f(x) = log3x y = log3x

Since we can not calculate logarithms with a base of 3, we will convert this from logarithmic notation to exponential notation.

y = log3x 3y = x

This function can be graphed now by finding coordinates to plot. Choose values for y to calculate the x-value.

x y

1 0 3 1 9 227 31/3 -11/9 -21/27 -3


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In previous sections, we discussed translations of functions. The graph of f(x)=log3x+2 is obtained by shifting f(x)=log3x up two units. Below are other examples of translations and reflections.

1. f(x)=log3x 2 is obtained by shifting f(x)=log3x down two units.2. f(x)=log3(x2) is obtained by shifting f(x)=log3x two units to the right. 3. f(x)=log3(x+2) is obtained by shifting f(x)=log3x two units to the left.4. f(x)=-log3x is obtained by reflecting f(x)=log3x across the through the x axis

Graphing a function does not always need to be obtained by translations. Picking points is always a good alternative. For this function, we will first need to change the function from logarithmic form to exponential form.

Then converting to logarithmic form, we get 3y–2 = x. We can then find points for this by choosing values for y.

y=log3x+2 can be written as y–2 = log3x by subtracting two on both sides.


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x y

1 2 3 3 9 4 27 51/3 11/9 01/27 -1

y=log3x+2 3y-2 = x

This function can be graphed now by finding coordinates to plot. Choose values for y to calculate the x-value.

Next Slide

Example 5. Graph f(x)=log3x+2


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3Graph f(x) log (x 4)

x y

-3 0 -1 1 5 2 23 3-3 2/3 -1-3 8/9 -2

y3

y

y log (x 4) 3 x 4

x 3 4

Answer:

The End.B.R.1-11-07

Let’s first observe this example with a base of 10.

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Transcript of Let’s first observe this example with a base of 10.