Box 6.0: Common Exponential and Log Identities


© 1999 by Karl Hahn
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Adding the Exponents: If b is any positive real number then

   bx by  =  bx+y
for all x and y. This is the single most important identity concerning logs and exponents. Since  ex  is only a special case of an exponential function, it is also true that
   ex ey  =  ex+y

Multiplying the Exponents: If b is any positive real number then

    (bx)y  =  bxy
 
for all x and y. Again since ex is a special case of an exponential function, it is also true that
    (ex)y  =  exy
 

Converting to roots to exponents: The nth root of x is the same as

   x1/n
for all positive x. Since square roots are a special case of nth roots, this means that
    _
   √x  =  x1/2
In addition:
    __
   √ex  =  ex/2

Converting to  ex  form: If b is any positive real number then

   bx  =  ex ln(b)
for all x. This includes the case where you have  xx
   xx  =  ex ln(x)
or if you have  f(x)x
   f(x)x  =  ex ln(f(x))
or if you have  xf(x)
   xf(x)  =  ef(x) ln(x)
or if you have  f(x)g(x)
   f(x)g(x)  =  eg(x) ln(f(x))
As an example, suppose you had  (x2 + 1)1/x.  That would be the same as
   e(1/x) ln(x2 + 1)

ex  is its own derivative: The derivative of  ex  is  ex.  This is the property that makes  ex  special among all other exponential functions.

ex  is always positive: You can put in any x, positive or negative, and  ex  will always be greater than zero. When x is positive,  ex > 1.  When x is negative,  ex < 1.  When  x = 0  then  ex = 1

The log of the product is the sum of the logs: Let b, x, and y all be positive real numbers. Then

   logb(xy)  =  logb(x) + logb(y)
This is the most important property of logs. Since  ln(x) = loge(x),  it is also true that
   loge(xy)  =  ln(xy)  =  loge(x) + loge(y)  =  ln(x) + ln(y)

The log of the reciprocal is the negative of the log: For any positive b, x, and y

   logb(1/x)  =  -logb(x)

   logb(y/x)  =  logb(y) - logb(x)
This includes
   ln(1/x)  =  -ln(x)

   ln(y/x)  =  ln(y) - ln(x)

Concerning multiplying a log by something else: Let b and x be positive and k any real number. Then

   k logb(x)  =  logb(xk)
This includes
   k ln(x)  =  ln(xk)
It also means that
         _
   logb(√x)  =  (1/2)logb(x)
and
       _
   ln(√x)  =  (1/2)ln(x)

Converting log bases to natural log You can compute any base log using the natural log function (that is ln) alone. If b and x are both positive then


   logb(x)  =

  ln(x)
       
  ln(b)

Every log function is the inverse of some exponential function: If b is any positive real number, then

   blogb(x)  =  logb(bx)  =  x
The right-hand part of this equation is true for all x. The left-hand part is true only for positive x. The functions,  ex  and  ln(x)  are also inverses of each other.
   eln(x)  =  ln(ex)  =  x
The same rules for x apply as above.

The derivative of the natural log is the reciprocal: If x is positive, it is always true that the derivative of  ln(x)  is  1/x

To find the derivative of logs of other bases, apply the conversion rule. So for the derivative of  logb(x)  you end up with

      1
          
   x ln(b)

The natural log can be expressed as a limit: For all positive x

                     xh - 1
   ln(x)  =   lim          
             h -> 0     h

You can only take the log of positive numbers: If x is negative or zero, you CAN'T take the log of x -- not the natural log or the log of any base. In addition, the base of a log must also be positive. As x approaches zero from above,  ln(x)  tends to minus infinity. As x goes to positive infinity, so does  ln(x).  So ln(x) has no limit as x goes to infinity or as x goes to zero.

Natural log is positive or negative depending upon whether x is greater than or less than 1: If  x > 1, then ln(x) > 0.  If  x < 1,  then  ln(x) < 0.  If  x = 1  then  ln(x) = 0.  Indeed the log to any base of 1 is always zero.


Something you Can't Do with Logs

There is no formula for the log of a sum: Don't go saying that log(a+b) is equal to log(a) log(b) because this is NOT TRUE.


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