Box 6.0: Common Exponential and Log IdentitiesKCT logo

© 1999 by Karl Hahn

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 
    _
   Öx  =  x1/2
In addition: 
    __
   Öex  =  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(Öx)  =  (1/2)logb(x)
and 
       _
   ln(Öx)  =  (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 

               ln(x)
   logb(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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