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Mathematics LibreTexts

0.4: Logarithms and Exponents

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Definition

The function y=ax is called an exponential function of base a and with exponent x provided that a>0,a1.

The function y=loga(x) is called logarithmic function with base a, provided that a>0 and a1,

and is defined by

y=loga(x) if and only if x=ay provided that a>0 and a1.

Rules for logarithms

Let a>0,a1.

  1. loga(x) is defined only when x>0.
  2. loga(xy)=loga(x)+loga(y)
  3. loga(xy)=loga(x)loga(y)
  4. nloga(x)=loga(xn)
  5. loga(1)=0
  6. loga(a)=1

Rules for Exponents:

Let a>0,a1.

 ax>0 for all real numbers x.

  1. ax+y=axay
  2. ax=1ax
  3. a0=1
  4.  
  5.  
  6.  

 

 

The transcendental number e is one of the most common bases employed. This notation was introduced in honor of the Swiss mathematician Euler (1707-1783). The definition of e which follows is certainly beyond the scope of these notes since it uses the concept of the limit. However, it is included here for completeness and as partial preparation for first calculus course.

Definition

e=limn(1+1n)n and it has been found that e2.71828 It has been found that scientific calculations involving exponential and logarithmic functions are simpler, if the base involved is e.

Natural Exponents and Natural Logarithms

 


This page titled 0.4: Logarithms and Exponents is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Pamini Thangarajah.

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