Exponents and Logarithms Study Guide

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:

a^l = 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, = 5^{4 – 4} = 5⁰. Thus, 5⁰ = 1. In general:

a⁰ = 1

Simplify the following.

3⁰ = 1

200⁰ = 1

The second consequence follows from:

while also = 2^3–7 = 2^–4. Thus, 2^–4 = . In general: a^–n =

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 e^x and the natural logarithm takes it right back to x, so ln(e^x ) = x. Similarly, e^1n(x) = x.

Example

The function f(x) = 2^x has the graph shown in Figure 3.1. Note that 2^x is quite different from x². For example, when x = 10, the value of 2^x 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) = 3^x 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 = e^x fits between y = 2^x and y = 3^x (see Figure 3.3).

Other than the strange base, everything about e^x is normal.

e⁰ = 1

eⁿ · e^m = e^n+m

e¹ = e

Another useful function is the opposite of e^x, 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 e^x.

If y = e^x, then ln(y) = ln( e^x) , so ln(y) = x.

The graph of y = ln(x) comes from flipping the graph of y = e^x 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 10^x = 7.

Solution 1

Take the natural logarithm of both sides.

ln(10^x) = 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.