CH 6 Overview: Exponential and Log Functions
What is the range of a log function?
(-∞, ₊∞)
What is the domain of a log function?
(0,∞)
THE CHANGE-OF-BASE FORMULA
A formula used to evaluate logs with bases other than "e" or "10"; either natural logs or logs with a base "10" may be used in this formula, as both yield the same answer
What is exponential growth?
A function that models exponential growth grows by a rate proportional to the amount present. For any real number, x, and any positive number real numbers a and b such that b does not equal 1, an exponential growth has a form f(x)=ab^x
natural logarithm
A logarithm with base e. It can be denoted log(e), but is more often denoted by ln.
By definition, an exponential function has a ___________________ as a base and an ________________________ __________________ as an exponent.
By definition, an exponential function has a constant as a base and an independent variable as an exponent.
What functions include exponential models?
Continuous growth and decay models
Domain and range of an logarithmic function
Domain: (-∞,+∞) Range: (0,+∞) Logarithmic functions and exponential functions are inverses of each other; therefore, the domain of a log function will be the range of an exponential funcion.
Domain and Range of exponential Parent Function
Domain: (0,+∞) Range: (-∞,+∞)
What is one of the most effective tools in math to understand what is occurring in the real world?
Graphs of functions
What does the result stand for?
How much you grew
Give an example of a logarithmic function.
In 2010, a major earthquake struck Haiti, destroying or damaging over 285,000 homes19. One year later, another, stronger earthquake devastated Honshu, Japan, destroying or damaging over 332,000 buildings,20 like those shown in Figure 1. Even though both caused substantial damage, the earthquake in 2011 was 100 times stronger than the earthquake in Haiti. How do we know? The magnitudes of earthquakes are measured on a scale known as the Richter Scale. The Haitian earthquake registered a 7.0 on the Richter Scale21 whereas the Japanese earthquake registered a 9.0.22 The Richter Scale is a base-ten logarithmic scale. In other words, an earthquake of magnitude 8 is not twice as great as an earthquake of magnitude 4. It is 108−4=104=10,000108−4=104=10,000 times as great! In this lesson, we will investigate the nature of the Richter Scale and the base-ten function upon which it depends.
Define the following for an exponential function: Initial Value: b-value: domain range: y- intercepts horizontal asymptotes
Initial Value: a; where a is non-zero b-value: Positive, Real Number b does not equal 1 domain: R; all real numbers range: When a>0; all positive real numbers When a<0; all negative real numbers y- intercepts:(0,a),because anything to the zero power is 1, and 1* a= a horizontal asymptotes: y=0
Is e rational or irrational?
Irrational
Is base, b, ever negative?
No, it is always positive
Percent change
Percent change refers to a change based on a percent of the original amount
List types of accounts that utilize compound interest.
Savings instruments in which earnings are continually reinvested, such as mutual funds and retirement accounts
What does the exponent stand for?
The amount of time you grew for
annual percentage rate (APR)
The annual percentage rate (APR) of an account, also called the nominal rate, is the yearly interest rate earned by an investment account.
What is the equation for a logarithmic function?
The base b logarithm of a number is the exponent by which we must raise b to get that number.
What does the base of a log stand for?
The growth rate
Define nominal .
The term nominal is used when the compounding occurs a number of times other than once per year. In fact, when interest is compounded more than once a year, the effective interest rate ends up being greater than the nominal rate! This is a powerful tool for investing.
How do we read the logarithmic function?
We read a logarithmic expression as, "The logarithm with base b of x is equal to, y," or, simplified, "log base b of x is y." We can also say, "b raised to the power of y is x," because logs are exponents.
Why do we limit b to be a positive number?
When we take the root of a negative number, it is undefined therefore we limit b to be positive.
What is the purpose of creating functions that model real- life functions?
Working with an equation that describes a real-world situation gives us a method for making predictions.
Is a logarithmic function one-to-one? What does this mean?
Yes, it passes the vertical line test; it has an inverse equation.
What is the inverse of a logarithmic function?
an exponential function; x=b^y
State what happens when the b- value in an exponential function is as follows: b>1 0<b<1
b>1: function grows at a rate proportional to its size 0<b<1: function decays at a rate proportional to its size
What real world examples are exponential functions utilized for?
finance, forensics, computer science, and most of the life sciences.
Explanation of exponential growth.
https://www.youtube.com/watch?v=0BSaMH4hINY
Logarithms explained
https://www.youtube.com/watch?v=ntBWrcbAhaY https://www.youtube.com/watch?v=l6sG1paOiJE
why cant logarithmic functions be negative?
https://youtu.be/8i4Ce251p1Y Any negative where you take, for example, the 1/2 power (which would be a square root, is undefined.
compound interest
interest earned on both the principal amount and any interest already earned
What is the notation for natural log?
ln(x)
Exponential Decay
refers to a decrease based on a constant multiplicative rate of change over equal increments of time, that is, a percent decrease of the original amount over time.
Exponential Growth
refers to an increase based on a constant multiplicative rate of change over equal increments of time, that is, a percent increase of the original amount over time.
Linear Growth
refers to the original value from the range increases by the same amount over equal increments found in the domain.
Exponential Growth
refers to the original value from the range increases by the same percentage over equal increments found in the domain.