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Logarithm

The logarithmic function is defined as the inverse of an exponential function.
 
If any number N is expressed in the form ax then the index ‘X’ is called the logarithm of the number N to the base ‘a’.
Thus, if N = ax
Then x = logaN.
 
Generally, logarithm of any number is calculated to the base 10. When run base is not mentioned, it should be taken as 10.
 
Restrictions with logarithm of any number For logarithm of any number to be defined, the number should be greater than zero and base should be positive and not equal to 1.
 
For logax to be defined, x > 0 and a > 0 as well a ≠ 1.
 
It can also be seen with the help of the graph of log x (given alongside).
 
Following observations can be made from this graph :
Description: 12-1.tif
  • Value of y can be negative for some value of x.
  • Value of x cannot be negative in any case.
  • For constant x, if base is lying in between 0 and 1, then log x becomes a decreasing function. Otherwise it is an increasing function.

Some Important Properties

In case of all the following properties the standard restrictions on logarithm will be used.
  1. loga(XY) = logaX + logaY
     
    If the restrictions given regarding log of any number are not used, then a good number of contradictions about the numbers can be seen. One of the examples of a similar nature is as follows:
     
    log (12) = log (− 4 × −3) = log (−4) + log(−3)
     
    Now on the left hand side, we have a defined value, but on the right hand side the value is not defined. It is due to the fact that loga(XY) = logaX + logaY is possible only if X > 0 and Y > 0.
  2. loga(X/Y) = logaX − logaY
  3. loga(Xk) = k logaX, loga Description: 1224.png= 1/k × logaX
  4. loga1 = 0 (As a0 + 1)
  5. logXX = 1
  6. logaX = 1/logxa
  7. Base change rule
     
    logaX = logbX/logba
     
    ⇒ logaX ÷ logba = logbX
  8. Description: 1233.png
  9. If a >1 and X a, logaX > 0
  10. If 0 < x < y then log x < log y (i.e., log x is an increasing function). In particular if x > 1 then log x > 0 and if 0 < x < 1 then log x < 0.
  11. If a > 1 and 0 < x < y then logax < loga y and ax < ay. If 0 < a < 1 and 0 < x < y then loga x > loga y and ax > ay.
  12. loga1 = 0
  13. Log of 0 and negative numbers is not defined.




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