(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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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 (15501617) 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, log_{b}x, 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,
FillingInTheBoxMethod of Computing Logarithms
Examples In Short Writing a logarithmic equation is just a funny way of writing an exponential equation: Using Technology To compute log_{b}x using technology, use the following formulas (see "ChangeofBase Formula" below for an explanation of the first one):
Common Logarithm, Natural Logarithm

Let f(x) = log_{2}x. 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:
Logarithmic Identities
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 online 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 halflife 
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 halflife as large samples of the same material, and the same is true for doubling time in the case of exponential growth.
Doubling Time and HalfLife
For a quantity undergoing exponential growth Q(t) = Q_{0}e^{kt},the growth constant k and doubling time t_{d} for Q are related by
For a quantity undergoing exponential decay Q(t) = Q_{0}e^{kt},the decay constant k and halflife t_{h} for Q are related by

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