Work on Exp/Logs The following slides will help you to review the topic of exponential and logarithmic functions from College Algebra. Also, I am introducing the rules for differentiating these functions (for base e) in preparation for Monday’s class.
Exponential and Logarithmic Functions Recall the basic forms for An exponential function with base b A logarithmic function with base b
Base e e is an irrational number An exponential function with base e A logarithmic function with base e
Exponential and Logarithmic Forms of an EQ An exponential equation with base b can be rewritten as a logarithmic equation as follows: SAY: b to the x power = c SAY: log base b of c = x Base = b Exponent = the logarithm = x
Exponential and Logarithmic Forms of an EQ Practice changing forms: Exponential form Logarithmic form
Exponential and Logarithmic Forms of an EQ Practice changing forms: Exponential form Logarithmic form
An Exponential Application An application with the natural exponential function involved interest compounded continuously: where A = amount, P = principal, r = rate, and t = time in years
Example Suppose we have $10,000 to invest and we find a bank that will compound continuously at a rate of 2.5%. How long will it take for our money will double? Do you remember how to do this? See next slide…
Example Use: P = $10,000 A = $20,000 r = 0.025, solve for t
Now solve the equation By hand: Divide by 10000: Change to a logarithmic equation Divide by 0.025: Get a calculator approximation:
It is its OWN derivative! To differentiate: Given a function of the form we find that its derivative is It is its OWN derivative!
How does this work if a = e? In general: Given a function of the form we find that its derivative is How does this work if a = e?
An example: Given a function of the form its derivative is given as IN WORDS: we repeat the exponential function then multiply by the derivative of the exponent
Next rule: Given a function of the form we find that its derivative is
Next rule: Given a function of the form we find that its derivative is
Let’s see the chain rule here Given the function its derivative is given as
In words: reciprocal derivative of the variable x of the variable expression expression