In this analysis of logarithm bases I will be discussing and explaining some of the properties and functions dependent on formulas which can be obtained by the use of logarithms. Simply, a logarithm is just an exponent. View the following diagram below in figure 1. An alternate explanation is that, the logarithm of a number x to a base b is just the exponent you put onto b to make the result equal x. For instance, since 5² = 25, we know that 2 (the exponent to the five) is the logarithm of 25 to base 5. Represented as, log5(25) = 2.

 

Figure 1.

In a general expression, when applying variables to replace numbers, if x = by, then we say that y is “the logarithm of x to the base b” or “the base b logarithm of x”. Represented as,

y = logb(x). Therefore, any exponential equation can be written as a log just by changing the x and y positions to manipulate the equation.

Another way to look at it is that the logbx function is defined as the inverse of the bx function. These two statements express that inverse relationship, showing how an exponential equation is equivalent to a logarithmic equation. View the diagrams in figure 2 and figure 3.

                                                        

                                                                                                                                        

     Figure 2.                                                        Figure 3.

Logarithms follow exponential rules being exponents themselves. A few general statements include that one can observe in any base, the logarithm of 1 is 0. View the diagram in figure 4. Up until this point the examples used, contained positive numbers. Can we find the logarithm of a negative number? View the diagram in figure 5. In addition, in any base, the logarithm of the base itself is 1. View the diagram in figure 6. Logarithms follow other rules such as inverse properties. View the diagram in figure 7.

Figure 4.                        

 Figure 5.                                                                                                                                                                                                                                                                                                                                        

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                                      Figure 7.

Figure 6.                                                

Can you take the logarithm of square roots? View figure 8 below.

                                                                                                                                                                                                                                                                                                        Figure 8.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        

Now, there are two basic logarithms common and natural. Beginning with common, the   10 of a number. That is, the  of 10 necessary to equal a given number. The common logarithm of x is written log x. For example, log 100 is 2 since 102 = 100. The base is understood to be ten. View the diagram in figure 9.  The ...

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