LOGARTHMIC FUNCTIONS

LOGARTHMIC FUNCTIONS

1. one log on ONE side

   • EXPONENTIAL LOGARITHMS
     • • • therfore, x=9
     • Exponential Laws

2. one log on EACH side

   • REMOVE logs and solve
     • log(x-2)=log5
       • x-2=5
         • therfore, x=7

3. Logarithms to the base of 10 are called common logarithms that is why when writing common logarithms you do not need to include the base number

   • for example log100 also means

4. if b is any number such that b>0 and b cannot =1 and x>0 then,

   • • is equivalent to
       • • exponential form
     • logarithmic form
   • Note that the requirement x>0 is really a result of the fact that we are also requiring b>0 .
     • We are raising a positive number to an exponent and so there is no way that the result can possibly be anything other than another positive number
       • It is very important to remember that we can’t take the logarithm of zero or a negative number.

5. Logarithmic Laws

6. Example: Evaluate

   • • the logarithm is the exponent (power) which you have to raise a base to make the given value
       • to begin to evaluate this log, we should let it =y
         • • this way, we the equation easier to solve when we have to raise the base and move the values around
             • • because 3x3x3x3=81, y=4
   • we can find out so much by just looking at a logarithmic equation such as domain, range, x-intercepts, y-intercepts, and equations of vertical and horizontal asymptotes. Yet it all depends on whether they exist or not. To do this we can also graph the logarithmic equation
     • for example, y=log (x-5)-8
       • to be able to look deeper into this equation, we should look at the left hand side then the right hand side.
         • we already know of vertical and horizontal translations due to previous chapters and therefore we can compare our function to the original function of y=log x
           • the -5 causes the a translation of 2 units to the right
           • the -8 causes a translation of 8 units down

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