Logarithms

Question 1

Restate in logarithmic form:

2⁴ = 16

Answer 1

log₂ 16 = 4

Question 2

Restate in exponential form:

log₂ 16 = 4

Answer 2

2⁴ = 16

Question 3

In plainspeak,
what do logarithms do?

Answer 3

They answer,
“What exponent do you have to raise one number to to get another number?”

“Logarithms are nothing but exponents.”

Question 4

What is the
formal definition of a logarithm?

Answer 4

log_b a = c ⇔ b^c = a

a: argument
b: base
c: exponent (or power)

Question 5

What are the
two special logarithms?

Answer 5

1. Common Logarithm.
   base: 10
   regular notation: log₁₀ X
   special notation: log X.
2. Natural Logarithm.
   base: e
   regular notation: log_e X
   special notation: ln X

Question 6

How can e be described as a limit?

Answer 6

lim (1 + 1/n)^n
n→∞

Question 7

What is
e
rounded to the
thousandth decimal place?

Answer 7

e ≈ 2.718

Question 8

How do you know that a

• *logarithmic expression** is
• *completely expanded?**

Answer 8

There are no
powers,
products, or
quotients
remaining in the arguments of the logarithms.

Question 9

What are the
four basic properties of logarithms?

Answer 9

1. Power Rule
2. Product Rule
3. Quotient Rule
4. Change of Base Formula

Question 10

Properties of logarithms:

log_b (xy) = ___

Answer 10

Logarithmic Product Rule:

log_b x + log_b y = ___

• The log of a product is equal to the
• *sum** of the
• log of its factors**.

Question 11

Properties of logarithms:

log_b x + log_b y = ___

Answer 11

Logarithmic Product Rule:

log_b (xy) = ___

• The log of a product is equal to the
• *sum** of the
• log of its factors**.

Question 12

Properties of logarithms:

log_b (a^z) = ___

Answer 12

Logarithmic Power Rule:

z•log_b a = ___

• The log of a power is equal to the
• *power** times the
• log of the base**.

Question 13

Properties of logarithms:

z•log_b a = ___

Answer 13

Logarithmic Power Rule:

log_b (a^z) = ___

• The log of a power is equal to the
• *power** times the
• log of the base**.

Question 14

Properties of logarithms:

log_b (x/y) = ___

Answer 14

Logarithmic Quotient Rule:

log_b x – log_b y = ___

The log of a quotient is equal to
the difference between the logs of the
numerator and the
denominator.

Question 15

Properties of logarithms:

log_b x – log_b y = ___

Answer 15

Logarithmic Quotient Rule:

log_b (x/y) = ___

The log of a quotient is equal to
the difference between the logs of the
numerator and the
denominator.

Question 16

How can you evaluate

log_b a?

Answer 16

The Change of Base Rule:

• *log_x a**
• *log_x b**

Proof:

• log_b a
• log_b a = c (defining terms)
• b^c = a (definition of logarithm)
• log_x (b^c) = log_x a (introduce log_x)
• c•log_x b = log_x a (the power rule)
• c = log_x a (divide both sides by log_x b)
  log_x b
• a = log_x a (substitution)
  log_x b

Question 17

• *log_x a** = ___
• *log_x b**

Answer 17

log_b a

(the change of base rule)

Question 18

• *1** = ___
• *log_b a**

Answer 18

Reciprocal Property:
log_a b

Question 19

• *log_b a**
• *x**

equals (two things)?

Answer 19

(1/x) • log_b a

and

log_b a^1/x

Question 20

Properties of logarithms:

log_b 1 = ___

Answer 20

0

because

b⁰ = 1 ⇔ log_b 1 = 0

Question 21

Properties of logarithms:

log_b b = ___

Answer 21

1

because

b¹ = b ⇔ log_b b = 1

Question 22

Properties of logarithms:

b^log_bx = ___

Answer 22

x

Proof:

• b^log_bx
• b^log_bx = y (defining terms)
• log_b y = log_b x (restate exponential as logarithm)
• y = x
• x = x (substitution)

Question 23

Properties of logarithms:

log_b (b^x) = ___

Answer 23

x

Proof:

• log_b (b^x​)
• x•log_b b (the power rule)
• x•1 (substitution)
• x