Exponential functions are those that have An example of an exponential function is
We can shift exponential function using the same patterns from before.
The half life of a radioactive substance in the time it takes for half of the substance to decay.
Caffeine has a half life of 5.7 hours in the human body. If you drink a Coca-Cola at noon, how much caffeine will be in your body when you go to bed? What do we need to know? What time do you go to bed? 32mg of caffeine per 12oz can.
The exponential function with base e are very useful in describing continuous growth and decay.
We know that in exponential functions the exponent is a variable. When we wish to solve for that variable we have two approaches we can take. One approach is to use a logarithm. We will learn about these in a later lesson. The second is to make use of a property called the Equality Property for Exponential Functions.
Basically, this states that if the bases are the same, then we can simply set the exponents equal. This property is quite useful when we are trying to solve equations involving exponential functions. Let’s try a few examples to see how it works.
(Since the bases are the same we simply set the exponents equal.) Here is another example for you to try: Example 1a:
How can we solve an equation when the bases are not the same?? Does anyone have an idea how we might approach this?
(our bases are now the same so simply set the exponents equal) Let’s try another one of these.
Guidelines: ◦ When solving an exponential equation, you must isolate the exponential (or write as one exponential on each side) before you take the log of both sides. ◦ When solving a logarithmic equation, you must isolate the logarithm (or write as a single log on each side of the equation) before you raise both sides using a base b.
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