Exponential Functions Logarithmic Functions

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Exponential Functions

There are a few special kinds of functions that are commonly used. We’re going to talk about exponential and logarithmic functions in this class, and there are also trigonometric functions (like f(x) = sin x) that we won’t cover. Exponential functions have their variable in the exponent with a constant base. x x For example, f(x) = 2 and f(x) = 10 are exponential functions. x

Let’s look at the exponential function f(x) = 2 . When we evaluate this function at whole number values, we get 2 multiplied by itself x times. 1 2 3 For example, f(1) = 2 = 2, f(2) = 2 = 22 = 4, and f(3) = 2 = 222 = 8.

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Example (cont.)

x

For other values of x, mathematicians have invented meanings for 2 . For example, we have already used negative exponents to mean “one over…”, -2 2 and f(-2) = 2 = 1/2 = 1/4. We also have let anything to the zero power be one, 0 so f(0) = 2 = 1.

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Practice Problems

For the same function f(x) = 2x, do the following. • Compute f(-3) and f(-1). • We now have found function values for x = -3, -2, -1, 0, 1, 2, and 3. Plot these points carefully and draw a nice smooth curve through them. In the quiz, you’ll be choosing between graphs that are subtly different, so do this carefully. Answers: 1) 1/8 and 1/2. 2) There is a graph of this function on Slide 6.

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More on f(x) = 2

x

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Another mathematical invention that we’ve been using with exponents is writing radicals with fractional exponents. 1/2 For example, f(1/2) = 2 = 1.414… (the square root of 2), 3/4 and f(3/4) = 2 = 1.681… (the fourth root of 2 cubed). 0.75 Of course, your calculator may know this last one as 2 . In any case, if you plot these two points, you will see that they lie on the smooth curve you drew in problem 2 on the previous slide. This is evidence that our interpretation of negative and fractional exponents is a good one. Remember that your calculator knows the values for all of the exponential functions. x The graph of f(x) = 2 is shown on the next screen.

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Graph of f(x) = 2

x

This graph is pretty typical of exponential functions. They always go through (0,1). This one approaches the x-axis towards the left and goes to infinity towards the right. Some may be backward from this and go to infinity on the left.

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Logarithmic Functions

When we have solved equations, we have relied on one basic trick very heavily. We got rid of things by “doing the opposite.” For example, to get rid of the “plus 3” in 2x + 3 = 7, we would subtract 3 from both sides. To get rid of the “times 2” in 2x = 4, we would divide by 2 on both sides. To get rid of the “square” in (x + 2)2 = 9, we would take the square root of both sides. The logarithmic functions are precisely those functions that “do the opposite” of the exponential functions.

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Cont.

For example, the log base 2 function g(x) = log2 x does exactly the opposite of x the exponential function f(x) = 2 . x In terms of an equation, suppose we have 2 = 7, and we want to solve for x. x We have 2 on the left side, and we want to “get rid of it”. We should “do the opposite,” that is, take the log base 2 of both sides. x log2 (2 ) = log2(7) x = log2(7) Now, if we had a log2-button on our calculators, we’d be done. We’ll say we’re done anyway, and call this the exact solution. We’ll talk about using our calculators later.

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Practice Problems

Find the exact solution to the following equations. x 1. 2 = 3. x 2. 5 = 2. (use log5) x 3. 10 = 3. Answers: 1) x = log2(3) 2) x = log5(2) 3) x = log10(3)

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Log Base 10, a.k.a. Common Logs

In problem 3, you should have gotten x = log10(3). You should have a log10-button on your calculator. It turns out that in any application of logs, you can get by with any one particular log function. For a long time, and in many situations, it was log10 that was used. Big tables with values for the log10 function were very common. I’m guessing that that’s why these logs are called common logs, and instead of writing log10(3), we usually just write log(3). Here, we drop the little 10, kind of like how we don’t write the little 2 in a square root.

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Cont.

The log10-button on your calculator probably has “LOG” or “log” on it. log10(3) = log(3) = .477121255 [[On your calculator: 3, log. ]] x You can check this in the equation 10 = 3. .477121255 10 = 3.000000002 on my calculator. The .000000002 is round-off error. We can use the log10-button on our calculator to compute any log. It turns out that log2(x) = log(x)/log(2). We had log2(7) before. We can compute this with log(7)/log(2) = 2.807354922 [[ 7, log, , 2, log, =.]] x This was the solution to the equation 2 = 7, so we can check this. 2.807354922 2 = 7. (Try this on your calculator!) Next Slide

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Example

There is a similar formula for any log, so it’s pretty easy to solve equations involving exponential functions. x Suppose we want to solve the equation 5 = 15. Take log5 of both sides x log5(5 ) = log5(15) x = log5(15) = log(15)/log(5) = 1.682606194. Rounded to 4 decimal places, this is 1.6826.

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Practice Problems

Solve each equation, and write your answer in decimal form rounded correctly to x 4 decimal places. 10 = 100. x 1. 10 = 20. x 2. 5 = 35. x 3. 2 = 20.73 x 4. 4 = 16. Answers: 1) x = 1.3010 2) x = 2.2091 3) x = 4.3736 4) x = 2

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More about logs

The graph of a log function looks just like the graph of the corresponding exponential function, but it’s turned over on its side. x The top green graph is 3 , and the bottom red one is log3(x).

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Cont.

If you read about the properties of logs in the optional textbook, or any book that talks about logs, you’ll find some basic rules about logs. Roughly, these say that multiplication, division, and exponentiation inside a log are equivalent to addition, subtraction, and multiplication outside. I won’t say more about these, but you should be aware that these properties of logs exist, and if you think about them, you might see that they correspond to properties of exponents, and by bringing the level of computation down a level, they can make complicated arithmetic computations easier.

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