Chapter 5 Logarithmic Functions Flashcards
What is a logarithm?
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
What is the relationship between logarithms and exponents?
They are inverses of each other such that
log(10^n) = n
10^(log n) = n
Simplify
log(ab)
log(a) + log(b)
Simplify
log(a/b)
log(a) - log(b)
Simplify
log(b^t)
t * log(b)
What is the natural logarithm?
It is a log that uses the “e” value as its base
Expressed as “ln” instead of “log”
Simplify
log(5x^2)
log(5) +2 * log(x)
What is the doubling time?
The time it takes for an exponentially growing function to double in value from its initial value.
What is half time?
It is the time it takes for an exponentially decaying function to reach half of its initial value.
What is the domain of log(x)?
All positive numbers
What is the range of log(x)?
All real numbers
What is the rate of change and concavity of log(x)?
Increasing and concave down
What is the relationship between the graphs of y = 10^x and y = log(x)?
As they are inverses of each other the values of the x and y axes are interchange between them.
Order of magnitude
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.
What does the following express?x —> a^+
It is expressing that x slides towards “a” from the right of the graph
Vertical asymptote Notation
What does the following express?
x —> a^-
It is expressing, in vertical asymptote notation, that x slides toward a from the left
What are log scales and what are they best used for?
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
What is it mean to have a “transformed” function?
That the values are in log scales
How would you fit an exponential function to data?
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
