Algebra Project

A . Most Striking Property of Logarithms

The most striking property of logarithms for me is

logb u /v logb u - logb v

This simplifies taking the logarithms of numbers , as any number can easily be expressed as a quotient of any two numbers . Because of this it is the most useful among the three logarithmic properties . However all the properties are useful in their own right , and when taken together , help expedite solving logarithmic problems

Examples of this property in use , in conjunction with other properties of logarithmic functions

log2

128 /16 log2 128 - log2 16 log2 27 - log2 24 7 - 4 3

The second property is very useful here in solving this as it simplifies the problem without having to resort to undergoing numerous calculations . By applying it in the second step , it allows for the equation to boil down to simple algebra . Furthermore , it helps solve the problem without needing to use a calculator , which in this case is hard to use given that the problem is not in base 10 , but in base 2

logb (x2 - 7x 10 ) - logb (x-5 logb [ (x2 - 5x 6 (x-5 )] logb [ (x - 5 (x - 2 (x-5 )] logb [ (x -2 )]

By applying the property in the second step , it allows for the drastic reduction of the problem . Without using the property , there is no way to simplify the problem . In solving it , one would have to go through numerous calculations before reaching the final answer

log10 1000 - log10 10 log10 (1000 /10 log10 (100 log10 (102 2

By using properties of logarithms , the computations are drastically lessened . The second property is used in reducing the algebraic equation , which by itself would need several computational steps before arriving at the final answer . On the other hand , logarithms with base 10 have easily determinable values , such as log1010 1 , log10100 log10102 2 , and so on . This idea is used to solve this problem

B . Graphing exponential functions of the form y b - ax

To graph an exponential function , as with any other function , several points need to be determined and plotted on a graph . It is imperative to understand the behavior of exponential functions in to extrapolate from a limited number of points a proper exponential graph

x -3 -2 .5 -2 -1 .5 -1 -0 .5 0 0 .5 1 1 .5 2 2 .5 3

y 0 .125 0177 0 .25 0 .354 0 .5 0707 1 1 .41 2 2 .83 4 5 .65 8

The most important property of exponential functions is that they give the set ratio of change in a given amount of time . To illustrate , start by graphing a simple exponential function of the form y b - ax . Let b 0 for simplicity and let a 2 , so y 2x . Table 1 gives a representative set of points for the graph

Table SEQ Table \ ARABIC 1

Plotting the graph gives

At the negative side of the x-axis , the graph...