Logarithmic form and exponential form, Mathematics

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Logarithmic form and exponential form ; We'll begin with b = 0 , b ≠ 1. Then we have

y= logb x          is equivalent to                  x= b y

The first one is called logarithmic form and the second is called the exponential form.  Remembering this is the key to evaluating logarithms. The number b, is base.

Example Without a calculator give the precise value of following logarithms.

(a) log2 16 

 (d) log9 (1/531441) 

(e) log 1/6 36 

Solution

To rapidly evaluate logarithms the simplest thing to do is to convert the logarithm to exponential form.  Hence, let's take a look at the first one.

(a) log2 16

Firstly, let's convert to exponential form.

log2 16 =?        is equivalent to            2? = 16

Hence, we're really asking two raised to what gives 16.  As 2 raised to 4 is 16 we get,

log2 16 = 4       because            24 =16

We'll not do the remaining parts in fairly this detail, however they were all worked in this way.

 (d) log 9(1/531441) = -6        because            9-6  = 1/96  =    1 /531441

 (e) log 1/6 36   = -2      because                        (1/6)-2=62=36

Special logarithms

There are a some special logarithms that arise in many places. These are following,

Natural logarithm

                                        ln x = loge x

This log is called as the natural logarithm

Common logarithm

                                        log x = log10 x

This log is called as the common logarithm

In the natural logarithm the base e is the similar number as in the natural exponential logarithm which we saw in the last section. Given is a sketch of both of these logarithms.

2244_Logarithmic  graph.png

From this graph we get some very nice properties of the natural logarithm which we will use several times in this and later Calculus courses.

ln x → ∞                  as  x → ∞

ln x → -∞             as  x → 0, x > 0


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