1. Section 6.4 Solving Logarithmic and Exponential Equations

2. Suppose you have $100 in an account paying 5% compounded annually.
Create an equation for the balance B after t years
When will the account be worth $200?

3. In the previous example we needed to solve for the input
Since exponential functions are 1-1, they have an inverse
The inverse of an exponential function is called the logarithmic function or log
In other words
If b = 10 we have the common log

4. Example
Rewrite the following expressions using logs

5. Logarithms are just exponents
Evaluate the following without a calculator by rewriting as an exponential equation:

6. Logarithms are inverses of exponential functions so
To see why this works rewrite the logarithm as an exponential equation
Also
To see why this works rewrite the exponential equation as a logarithmic equation
Evaluate

7. Properties of the Logarithmic Function
Now the log we have in our calculator is the common log so b = 10. There is also the natural log, ln on our calculator where b = e. It has all the same properties.

8. The Natural Logarithm

9. Evaluate without a calculator
Simplify without a calculator

10. Change of Base Formula
It turns out we can write a log of a base as a ratio of logs of the same base
This is useful if our solution contains a log that does not have a base of 10 or e
The Change of Base Formula is
For us we typically use 10 or e for a since that is what we have in our calculator

11. Reminder: The half-life of a substance is the amount of time it takes for a decreasing exponential function to decay to half of its initial value
The half-life of iodine-123 is about 13 hours. You begin with 100 grams of iodine-123.
Write an equation that gives the amount of iodine remaining after t hours
Hint: You need to find your rate using the half-life information
Determine the number of hours for your sample to decay to 10 grams

12. Reminder: Doubling time is the amount of time it takes for an increasing exponential function to grow to twice its previous level
What is the doubling time of an account that pays 4.5% compounded annually? Quarterly?
Recall that the population of Phoenix went up by 45.3% between Assuming that growth remained steady, what is the doubling time of the Phoenix population?

13. Any exponential function can be written as Q = abt or Q = aekt
Then b = ekt so k = lnb
Convert the function Q = 5(1.2)t into the form Q = aekt
What is the annual growth rate?
What is the continuous growth rate?
Convert the function Q = 10(0.81)t into the form Q = aekt
What is the annual decay rate?
What is the continuous decay rate?

14. Let’s try a few from the chapter
6.4 – 1, 3, 5, 7, 9, 19, 27, 29, 37, 49