(This topic is also in Section 2.3 in Applied Calculus and Section 10.3 in Finite Mathematics and Applied Calculus)
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|Function Evaluator & Grapher||Excel Grapher||Java Grapher|
|Using and Deriving Algebraic Properties of Logarithms|
Brush up on your Algebra For all of the tutorials on nonlinear functions and models, you should be familiar with the algebra of exponents and radicals.Logarithms were invented by John Napier (1550-1617) in the late 16th century as a means of aiding calculation. Although computers and calculators have done away with that use of logarithms, many other uses remain. In particular, the logarithm is used to to model real world phenomena in numerous fields, including physics, finance, and economics.
|Base b Logarithm
The base b logarithm of x, logbx, is the power to which we need to raise the positive number b in order to get x.
For instance, the power to which we need to raise 2 in order to get 8 is 3. Therefore,
Filling-In-The-Box-Method of Computing Logarithms
In Short Writing a logarithmic equation is just a funny way of writing an exponential equation:
To compute logbx using technology, use the following formulas (see "Change-of-Base Formula" below for an explanation of the first one):
Common Logarithm, Natural Logarithm
Let f(x) = log2x. Fill in the missing values in the following table, and then click on the correct graph.
One thing that makes logarithms useful for solving equations is their properties:
The following identities hold for all positive bases a &ne 1 and b &ne 1, all positive numbers x and y, and every real number r. How they are derived is discussed in the on-line discussion "Using and Deriving Algebraic Properties of Logarithms."
Q Multiple Choice: ln(1/2) =
Q Multiple Choice: log(18) =
Q Multiple Choice: log(0.125) =
Q Multiple Choice: ln[3(1.01-2t)] =
|ln(3) - 2t ln(1.01)||-2t(ln(3) + ln(1.01))||2t(ln(3) - ln(1.01))|
|ln(3) + ln(1.01) - ln(2) - ln(t)||ln(3) - ln(2t) ln(1.01)||ln(3) - [ln(2) - ln(t) ln(1.01)]|
|Relationship with Exponential Functions
The following identities show that the operations of taking the base b logarithm and raising b to a power are "inverse" to each other:
|Exponential Growth & Decay
An exponential growth function has the form
Similarly, an exponential decay function has the form
Of interest here are the following questions:
|If a quantity is undergoing exponential growth, how long does it take for the quantity to double its original size? This time is called the doubling time.|
|If a quantity is undergoing exponential decay, how long does it take for the quantity to decay to half its original size? This time is called the half-life|
It turns out that the answers to these questions are independent of the size of the sample: small samples undergoing exponential decay have the same half-life as large samples of the same material, and the same is true for doubling time in the case of exponential growth.
|Doubling Time and Half-Life
For a quantity undergoing exponential growth Q(t) = Q0ekt,the growth constant k and doubling time td for Q are related by
For a quantity undergoing exponential decay Q(t) = Q0e-kt,the decay constant k and half-life th for Q are related by
You now have several options
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