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# Exponents and Logarithms Study Guide

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## Exponents

Because exponents form such an important part of calculus, we shall briefly review them. Generally, an means "multiply the base a as many times as the exponent n."

Note: The exponent formulas in this lesson all assume that a is a positive number.

#### Examples 1

Review the following examples by multiplying out.

34 = 3 · 3 · 3 · 3 = 81

25 = 2 · 2 · 2 · 2 · 2 = 32

51 = 5

106 = 1,000,000

A number to the first power is just that number:

al = a

When two numbers with the same base are multiplied, their exponents are added.

#### Examples 2

Review and simplify the following.

410 · 47 = 417

102 · 105 = 107

53 · 5 = 53 · 51 = 54

72 · 74 · 73 = 79

The rule about adding exponents has an interesting consequence. We know that √5 · √5 = 5 because this is what "square root" means. Also, however . Because √5 and act exactly the same, they are equal: √5 = . This works for square roots, cube roots, and so on:

#### Examples 3

Simplify the following.

When two numbers with the same base are divided, their exponents are subtracted.

#### Examples 4

Work through the following simplifications.

The rule about subtracting exponents has two interesting consequences. First,  = 1 because any nonzero number divided by itself is one. Also, = 54 – 4 = 50. Thus, 50 = 1. In general:

a0 = 1

Simplify the following.

30 = 1

2000 = 1

The second consequence follows from:

while also = 23–7 = 2–4. Thus, 2–4 = . In general: an =

#### Examples 5

Work through the following simplifications.

3–2 =

4–1 =

## Exponential Functions

We can form an exponential function by leaving the base fixed and varying the exponent.

### Exponents and Logarithms

The exponential function takes x to ex and the natural logarithm takes it right back to x, so ln(ex ) = x. Similarly, e1n(x) = x.

#### Example

The function f(x) = 2x has the graph shown in Figure 3.1. Note that 2x is quite different from x2. For example, when x = 10, the value of 2x is 210 = 2·2·2·2·2·2·2·2·2·2 = 1,024, while the value of x2 is 102 = 10· 10 = 100.

#### Example

The function g(x) = 3x has the graph shown in Figure 3.2. For reasons that will become clear later, a very nice base to use is the number e = 2.71828… , which, just like π = 3.14159… , can never be written out completely.

Because 2 < e < 3, the graph of y = ex fits between y = 2x and y = 3x (see Figure 3.3).

Other than the strange base, everything about ex is normal.

e0 = 1

en · em = en+m

e1 = e

Another useful function is the opposite of ex, known as the natural logarithm ln(x). Just as subtracting undoes adding, dividing undoes multiplying, and taking a square root undoes squaring, the natural logarithm undoes ex.

If y = ex, then ln(y) = ln( ex) , so ln(y) = x.

The graph of y = ln(x) comes from flipping the graph of y = ex across the line y = x, as depicted in Figure 3.4.

The laws of ln(x) are rather unusual.

ln( a) + ln(b) = ln( a · b)

ln(an) = n · ln(a)

The last of the three preceding laws is useful for turning an exponent into a matter of multiplication.

#### Example 1

Solve for x when 10x = 7.

#### Solution 1

Take the natural logarithm of both sides.

ln(10x) = ln(7)

Use ln(an) = n · ln(a).

x · ln(10) = ln(7)

Divide both sides by ln(10).

A calculator can be used to find a decimal approximation: ≈ 0.84509, If desired.

#### Example 2

Simplify ln(25) + ln(4) – ln(2).

#### Solution 2

Use ln(a) + ln(b) = ln(a · b).

ln(25 · 4) – ln(2)

Use .

Find practice problems and solutions for these concepts at Exponents and Logarithms Practice Questions.

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