Chapter 5 Logarithmic Functions

Question 1

What is a logarithm?

Answer 1

They calculate the exponent that is required for its base( typically 10 or e) to reach the value in parenthesis

Using Base 10
Log(100) = 2

Question 2

What is the relationship between logarithms and exponents?

Answer 2

They are inverses of each other such that

log(10^n) = n
10^(log n) = n

Question 3

Simplify

log(ab)

Answer 3

log(a) + log(b)

Question 4

Simplify

log(a/b)

Answer 4

log(a) - log(b)

Question 5

Simplify

log(b^t)

Answer 5

t * log(b)

Question 6

What is the natural logarithm?

Answer 6

It is a log that uses the “e” value as its base

Expressed as “ln” instead of “log”

Question 7

Simplify

log(5x^2)

Answer 7

log(5) +2 * log(x)

Question 8

What is the doubling time?

Answer 8

The time it takes for an exponentially growing function to double in value from its initial value.

Question 9

What is half time?

Answer 9

It is the time it takes for an exponentially decaying function to reach half of its initial value.

Question 10

What is the domain of log(x)?

Answer 10

All positive numbers

Question 11

What is the range of log(x)?

Answer 11

All real numbers

Question 12

What is the rate of change and concavity of log(x)?

Answer 12

Increasing and concave down

Question 13

What is the relationship between the graphs of y = 10^x and y = log(x)?

Answer 13

As they are inverses of each other the values of the x and y axes are interchange between them.

Question 14

Order of magnitude

Answer 14

If one object is 10x heavier than another, we say it is an order of magnitude heavier.
With two factors of 10x, which is 10^2, then two orders of magnitude.

Question 15

What does the following express?x —> a^+

Answer 15

It is expressing that x slides towards “a” from the right of the graph

Vertical asymptote Notation

Question 16

What does the following express?

x —> a^-

Answer 16

It is expressing, in vertical asymptote notation, that x slides toward a from the left

Question 17

What are log scales and what are they best used for?

Answer 17

Log scales are the smaller numbers that are derived when calculating the log of larger numbers

Used to reduce the variation between different numbers in a data set

Question 18

What is it mean to have a “transformed” function?

Answer 18

That the values are in log scales

Question 19

How would you fit an exponential function to data?

Answer 19

Transform the relationship by taking the natural log of both sides and substituting y with ln(y)

ln(y)=ln(ae^(kt))

Simplify to ln(y) = ln(a) + kt

Use linear regression on the variables y and t

Transform the linear regression equation back into our original variables by solving for Y