# Inverses of Logarithms and Exponents Worksheets

Logarithms and exponents are arc enemies, well not really. They are the exact inverse of one another meaning that if you have a logarithmic function and combine it with an equivalent exponential function, they will cancel one another out. Learning how to apply this unique relationship to everyday situations that have been modelled with functions is a form of discrete mathematics that has been known to lead people down a path toward very lucrative careers. This is a series of worksheets and lessons that show students how logs undo exponentials and the relationship that exists between those two.

### Aligned Standard: HSF-BF.B.5

- Sneaker Company Growth Step-by-step Lesson- The shoe company wants to grow six times their current profit. How long will it take?
- Guided Lesson - How long will it take to pay off the mortgage and "Hello!" Richter scale.
- Guided Lesson Explanation - The Richter scale problem will give many students some trouble.
- Practice Worksheet - We stay on the same themes: paying loans back and fixed growth scales.
- Matching Worksheet - I might have forgot a dollar sign or two to make the problems not so obvious.

- Answer Keys - These are for all the unlocked materials above.

### Homework Sheets

In some cases, the companies presented have an outrageous profit growth. They actually were modelled after real companies.

- Homework 1 - The profit growth of a tea company is 8% every 2 years. If the company were to stay on that track, how long would it take for the company to grow to 5 times its current profit?
- Homework 2 - Find the monthly payments on a $200,000, 23-year mortgage, with monthly payments at 12%. How much interest will you pay over the 23 years?
- Homework 3 - Early in the century, an earthquake with the magnitude of 4 struck in the USA. During the same year, Japan recorded an earthquake that was three times greater. What was the magnitude of the earthquake in Japan?

### Practice Worksheets

Hello, Richter scale!

- Practice 1 - The Richter scale is used to measure the intensity of Earthquakes. He Richter scale is based on the equation: R= log (I ÷ I0)
- Practice 2 - Go over the three fundamental methods that apply to these.
- Practice 3 - An earthquake with the magnitude of 5 struck in Russia. During the same year, Africa recorded an earthquake that was three times greater. What was the magnitude of the earthquake in Africa?

### Math Skill Quizzes

We mix the two skills together here.

- Quiz 1 - We sum it all up on the quiz.
- Quiz 2 - We can calculate the growth potential of the company by using the growth formula.
- Quiz 3 - Substitute the information given into formula.

### The Relationship of Logarithmic and Exponential Functions

Just like their equational counterparts logarithmic functions are just functions that somewhere within the function itself contain a logarithm. Exponential functions are those that have the presences of a exponent found within it. Logarithms and exponential functions hold an important place in the concept of mathematics. They have an inverse relationship with each other. Meaning that you can use one to undo the other. They are the exact opposites of one another.

Exponential Functions can be written as f(x) = e^{x}, where the value of e is 2.718. This function is used when the value of x is fairly complicated. On the other hand, logarithms that are in a counterpart relationship with the exponential function is called the natural log. The natural log (ln) is the inverse function of the exponential function. In other words, the graph of ln is a reflection across the line with the exponential function's graph.

When see a function stated with an exponent of -1 you are seeing the inverse of that function stated. So f^{-1}(x) is the inverse of f(x). So we can use this relationship between logarithmic and exponential functions to manipulate them in more complex situations. This can be helpful for modelling just about anything that is growing or decaying at exponential rates. This type of math lends itself to many different financial and scientific applications that make a big difference in the everyday world.