Chapter 5: Exponential and Logarithmic Function

Definition of Logarithms

Suppose b>0 and b≠ 𝟏. 𝐅𝐨𝐫 𝒏 >𝟎, 𝐭𝐡𝐞𝐫 𝐢𝐬 𝐧𝐮𝐦𝐛𝐞𝐫 𝒑 𝐬𝐮𝐜𝐡 𝐭𝐡𝐚𝐭 𝒍𝒐𝒈𝒍𝒐𝒈𝒃𝒏 =𝒑 𝐢𝐟 𝐚𝐧𝐟 𝐨𝐧𝐥𝐲 𝐢𝐟 𝒃𝒑 = 𝒏

Exponential Functions: The "Natural" Exponential "e"

In this section we consider these functions.

Name General Form

Linear

Quadratic

Cubic

Absolute value

Exponential

Logarithmic

Reciprocal

𝑓(𝑥) = 𝑎𝑥 + 𝑏,𝑎 ≠ 0

𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐,𝑎 ≠ 0

𝑓(𝑥) = 𝑎𝑥 3 + 𝑏𝑥 2 + 𝑐𝑥 + 𝑑, 𝑎≠ 0

𝑓(𝑥) = |𝑥|

𝑓(𝑥) = 𝑎𝑥 , 𝑎 > 0, 𝑎 ≠ 1

𝑓(𝑥) = 𝑙𝑜𝑔𝑒𝑥 𝑜𝑟 𝑓(𝑥) = 𝐼𝑛 𝑥

𝑓(𝑥) = 𝑘

𝑥, 𝑥 ≠ 0

f : x ↦ 𝑎𝑥 + 𝑏, 𝑎 ≠ 0

𝑓: 𝑥 ↦ 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 𝑎 ≠ 0

𝑓: 𝑥 ↦ 𝑎𝑥 3 + 𝑏𝑥 2 + 𝑐𝑥 + 𝑑, 𝑎 ≠ 0

𝑓: 𝑥 ↦ |𝑥|

𝑓: 𝑥 ↦ 𝑎𝑥 , 𝑎 < 0

𝑓: 𝑥 ↦ 𝐼𝑛 𝑥

𝑓: 𝑥 ↦𝑘

𝑥, 𝑥 ≠ 0

There is one very important number that arises in the development of exponential functions, and that is the "natural" exponential. (If you really want to know about this number, you can read the book "e: The Story of a Number", by Eli Maor.)

In the previous page's discussion of compound interest, recall that "n" stood for the number of compoundings in a year. What happens when you start compounding more and more frequently? What happens when you go from yearly to monthly to weekly to daily to hourly to minute-ly to second-ly to...?

Ignoring the principal, the interest rate, and the number of years by setting all these variables equal to "1", and looking only at the influence of the number of compounding, we get:

how often compounded

computation

yearly

semi-annually

quarterly

monthly

weekly

daily

hourly

every minute

every second

As you can see, the computed value keeps getting larger and larger, the more often you compound. But the growth is slowing down; as the number of compounding increases, the computed value appears to be approaching some fixed value. You might think that the value of the compound-interest formula is getting closer and closer to a number that starts out "2.71828". And you'd be right; the number we're approaching is called "e".

If you think back to geometry, you'll remember the number "pi", which was approximated by the decimal "3.14159" or the fraction " 22/7 ". Remember that we call pi by the name "pi" and use a symbol for this number because pi never ends when written as a decimal. It's not a "neat" number like 2 or –1/3; it is in fact an irrational number. But it's an important number; you'd have real trouble doing geometry without it. So we give this useful number the name "pi", to simplify our calculations and communication, because it's a lot easier to say "pi" than to say "3.141592653589 and so on forever" every time we need to refer to this number. We gave the number a letter -name because that was easier.

In the same way, this compound-interest number is also very useful. You may not see the usefulness of it yet, but it is vital in physics and other sciences, and you can't do calculus without it. As with pi, listing out its first dozen or so digits every time we refer to this number gets to be annoying, so we call it by the name "e".

The number "e" is the "natural" exponential, because it arises naturally in math and the physical sciences (that is, in "real life" situations), just as pi arises naturally in geometry. This number was discovered by a guy named Euler (pronounced "OY-ler"; I think he was Swiss), who described the number and named the number "e", and then swore that this stood for "exponential", and not for his own name.

Your calculator can do computations with e; it is probably a "second function" on your calculator, right above the "ln" or "LN" key on your calculator.

Given f(x) = ex, evaluate f(3), rounding to two decimal places.

I need to plug this into my calculator. (Check your owner's manual, if you're not sure of the key sequence.) I get: Copyright © Elizabeth Stapel 2002-2011 All Rights Reserved

f(3) = 20.0855369232...

Rounded to two decimal places, the answer is f(3) = 20.09.

Graph y = e2x.

Since e is greater than 1, and since "2x" is "positive", then this should look like exponential growth. I will compute some plot points:

Then I'll draw the graph:

Make sure, when you are evaluating e2x, that you format the expression correctly. Either multiply out the "2x" first, and then apply it to the e, or else put the "2x" inside parentheses. Otherwise, the calculator will think you mean "e2 × x", and will return the wrong values, as is demonstrated at right:

Your teacher or book may go on at length about using other bases for growth and decay equations, but, in "real life" (such as physics), the natural base e is generally used. The equation for "continual" growth (or decay) is A = Pert, where "A", is the ending amount, "P" is the beginning amount (principal, in the case of money), "r" is the growth or decay rate (expressed as a decimal), and "t" is the time (in whatever unit was used on the growth/decay rate). Make sure you have memorized this equation, along with the meanings of all the variables. You are almost certain to see it again, especially if you are taking any classes in the sciences.

(This equation helped me pass a chemistry class. I really didn't know what the teacher was talking about, but all the test problems worked off this equation, so I just plugged in all the given information, and solved for whichever variable was left. I'm not saying this to advocate being clueless in chemistry, but to demonstrate that the above really is a useful equation.)

The continuous-growth formula is first given in the above form "A = Pert", using "r" for the growth rate, but will later probably be given as A = Pekt, where "k" replaces "r", and stands for "growth (or decay) constant". Or different variables may be used, such as Q = Nekt, where "N" stands for the beginning amount and "Q" stands for the ending amount. The point is that, regardless of the letters used, the formula remains the same. And you

should be familiar enough with the formula to recognize it, no matter what letters happen to be included within it.

Certain bacteria, given favorable growth conditions, grow continuously at a rate of4.6% a day. Find the bacterial population after thirty-six hours, if the initial population was 250 bacteria.

As soon as I read "continuously", I should be thinking "continuously-compounded growth formula". "Continuously" is the buzz-word that tells me to use "A = Pert". The beginning amount was P = 250, the growth rate is r = 0.046, and the time t is 36/24 = 1.5 days.

Why is "time" converted to days this time, instead of to years? Because the growth rate was expressed in terms of a given percentage per day. The rates in the compound-interest formula for money are always annual rates, which is why t was always in years in that context. But this is not the case for the general continual-growth/decay formula; the growth/decay rates in other, non-monetary, contexts might be measured in minutes, hours, days, etc.

I plug in the known values, and simplify for the answer:

A = 250e(0.046)(1.5) = 250e(0.069) = 267.859052287...

There will be about 268 bacteria after thirty-six hours.

By the way, if you do your calculations "inside-out", instead of left-to-right, you will be able to keep everything inside the calculator, and thereby avoid round-off error.

For example, the above computation would be done like this:

Exponential Function

Exponential functions look somewhat similar to functions you have seen before, in that they involve exponents, but there is a big difference, in that the variable is now the power,

rather than the base. Previously, you have dealt with such functions as f(x) = x2, where the variable x was the base and the number 2 was the power. In the case of exponentials, however, you will be dealing with functions such as g(x) = 2x, where the base is the fixed number, and the power is the variable.

Let's look more closely at the function g(x) = 2x. To evaluate this function, we operate as usual, picking values of x, plugging them in, and simplifying for the answers. But to evaluate 2x, we need to remember how exponents work. In particular, we need to remember that negative exponents mean "put the base on the other side of the fraction line".

So, while positive x-values give us values like these:

...negative x-values give us values like these:

Copyright © Elizabeth Stapel 2002-2011 All Rights Reserved

Putting together the "reasonable" (nicely graphable) points, this is our T-chart:

...and this is our graph:

Logarithmic Function

Logarithms are the "opposite" of exponentials, just as subtraction is the opposite of addition and division is the opposite of multiplication. Logs "undo" exponentials. Technically speaking, logs are the inversesof exponentials.

In practical terms, I have found it useful to think of logs in terms of The Relationship:

—The Relationship—

y = bx

..............is equivalent to............... (means the exact same thing as)

Logb(y) = x

On the left-hand side above is the exponential statement "y = bx". On the right-hand side above, "logb(y) = x" is the equivalent logarithmic statement, which is pronounced "log-base-b of y equals x"; The value of the subscripted "b" is "the base of the logarithm", just as b is the base in the exponential expression "bx". And, just as the base b in an exponential is always positive and not equal to 1, so also the base b for a logarithm is always positive and not equal to 1. Whatever is inside the logarithm is called the "argument" of the log. Note that the base in both the exponential equation and the log equation (above) is "b", but that the x and y switch sides when you switch between the two equationsIf you can remember this relationship (that whatever had been the argument of the log becomes the "equals" and whatever had been the "equals" becomes the exponent in the exponential, and vice versa), then you shouldn't have too much trouble with logarithms.

(I coined the term "The Relationship" myself. You will not find it in your text, and your teachers and tutors will have no idea what you're talking about if you mention it to them. "The Relationship" is entirely non-standard terminology. Why do I use it anyway? Because it works.)

By the way: If you noticed that I switched the variables between the two boxes displaying "The Relationship", you've got a sharp eye. I did that on purpose, to stress that the point is not the variables themselves, but how they move.

Convert "63 = 216" to the equivalent logarithmic expression.

To convert, the base (that is, the 6)remains the same, but the 3 and the 216 switch sides. This gives me:

log6(216) = 3

Convert "log4(1024) = 5" to the equivalent exponential expression.

To convert, the base (that is, the 4) remains the same, but the 1024 and the 5 switch sides. This gives me:

45 = 102

Law of Logarithms

Since a logarithm is simply an exponent which is just being written down on the line, we expect the logarithm laws to work the same as the rules for exponents, and luckily, they do.

3 important LAW OF LOGARITHMS

log A + logB =log(AB)

log A –logB =log𝑨

𝑩, 𝑩 ≠ 𝟎

𝐥𝐨𝐠 𝑨 = 𝐥𝐨𝐠 𝑨𝒏

Exponents Logarithms

`b^m × b^n = b^(m+n)` ` log_b xy = log_b x + log_b y`

`b^m ÷ b^n = b^(m-n)` `log_b (x/y) = log_b x − log_b y`

`(b^m)^n = b^(mn)` `log_b (x^n) = n log_b x`

`b^1 = b` `log_b (b) = 1`

`b^0 = 1` `log_b (1) = 0`

Note: On our calculators, "log" (without any base) is taken to mean "log base 10". So, for example "log 7" means "log107".

Examples 1. Expand

log 7x

as the sum of 2 logarithms.

Using the first law given above, our answer is

`log\ 7x = log\ 7 + log\ x`

Note 1: This has the same meaning as `10^7 xx 10^x = 10^(7+x)`

Note 2: This question is not the same as `log_7 x`, which means "log of x to the base `7`", which is quite different.

2. Using your calculator, show that

`log (20/5) = log\ 20 − log\ 5`.

I am using numbers this time so you can convince yourself that the log law works.

LHS

`= log (20/5)`

`= log\ 4 ̀

`= 0.60206` (using calculator)

Now

RHS

`= log\ 20 − log\ 5`

`= 1.30103 − 0.69897` (using calculator)

`= 0.60206 `

= LHS

.

3. Express as a multiple of logarithms: log x5.

Using the third logarithm law, we have

`log\ x^5 = 5\ log\ x`

We have expressed it as a multiple of a logarithm, and it no longer involves an exponent.

Note 1: Each of the following is equal to 1:

log6 6 = log10 10 = logx x = loga a = 1

The equivalent statements, using ordinary exponents, are as follows:

61 = 6

101 = 10

x1 = x

a1 = a

Note 2: All of the following are equivalent to `0`:

log7 1 = log10 1 = loge1 = logx 1 = 0

The equivalent statments in exponential form are:

70 = 1

100 = 1

e0 = 1

x0 = 1

Examples: 1. Express as a sum, difference, or multiple of logarithms:

`log_3((root(3)y)/8)`

`log_3((root(3)y)/8)`

` =log_3(root(3)y)-log_3(8)`

`=log_3(y^(1//3))-log_3(2^3)`

`=1/3log_3(y)-3\ log_3(2)`

2. Express

2 loge 2 + 3 loge n

as the logarithm of a single quantity.

Applying the logarithm laws, we have:

2 loge 2 + 3 loge n

= loge 4 + loge n3

= loge 4n3

Note: The logarithm to base e is a very important logarithm. You will meet it first in Natural Logs (Base e) and will see it throughout the calculus chapters later.

3. Determine the exact value of:

`log_3root(4)27`

Let

`log_3root(4)27=x`

Then

`3^x=root(4)27`

Now

`root(4)27=root(4)(3^3)=3^(3//4)`

So x = 3/4.

Therefore

`log_3root(4)27=3//4`

4. Solve for y in terms of x:

log2x + log2y = 1

Using the first log law, we can write:

log2 xy = 1

Then xy = 21

So

`y=2/x`

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