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Introductory Calculus: Exponentials and Logarithms
online assistance Properties of Exponents: To review some of the fundamental properties of exponents, we will look at several examples.
If a > 0, b > 0, and x and y are any real numbers, then:
  1. axay = ax+y
online assistance Properties of Logarithms: To review some of the fundamental properties of logarithms, we will look at several examples.
If x and y are both positive numbers, then:
  1. Meaning of Log: If .
Notice: the log is the exponent.
log(x) + log(x + 2) = log(x2 + 2x)
Notice: if the base b is not shown, the base is assumed to be 10.
  1. ln(x) means loge(x) where .
Putting It All Together:
Solve for x if .
By direction inspection we can see that x must equal 4.
Solve for x if .
In this case, direct inspection fails us since 4 is too big - giving us 16 - and 3 is too small - only giving us 8.
To get an exact answer, we would use logarithms:
Step 1:
Step 2:
Step 3:

Let's Practice. In the following five equations, solve for x (accurate to 3 decimal places).

Graphs: A graph of both is shown below right. Notice that in order to graph , it is necessary to rewrite as since there is no log base 2 button on the calculator. Also, notice that the point (1, 2) is on . Since is its inverse, the point (2, 1) will be on that graph.
Derivatives: Given below are ten examples of taking derivatives of exponential functions and logarithms.


  2. Show that has a relative minimum at .

We know from example #3 above that .
If we set this equal to 0, we get .
We note that for , this derivative is negative, and for , the derivative is positive.
This change in sign from negative to positive is the proof that we need to know that has a minimum value when .
Let's Practice.
Using example #8 above as a guide, show that has a relative minimum at x = 1.  Be careful to check values of very close to 1 for the sign of .

Using example #7 above as a guide, find any max or min of .

M Ransom

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