Logs Theory of Logarithms s

1. Logs Theory of Logarithmss

2. The Exponential Function y=ax Base power or index For example: y=2x So if x = 2 Then y= 4 Note: Exponential functions are always positive i.e. above the x-axis.

3. Exponential Equations Sometime we have to solve equations involving an exponential function. For example : 8=2x We might be able to guess the answer but we could also use the graph starting on the y=axis and tracing back to the x-axis. 8 3

4. The Logarithm Function To make solving exponential equations easier we can reverse the x and y axes of the exponential function. + + Rotate Reflect + + + + This new function is called the logarithm function and is written y=log a x Read this as “ y= log to the base a of x”

5. The Logarithm Examples The following pairs of statements are equivalent 8=23 3=log 2 8 81=34 4=log 3 81 625=54 4=log 5 625 Remember: The log of a number is a power of the base .

6. The Laws of Logs – Multiplication Law Supposep=log a n andq=log a m Thenap=nandaq=m Thereforenm= apaq nm= ap+q (Law of index addition) p+q= log a nm log a n + log a m= log a nm

7. The Laws of Logs – Quotient Law Supposep=log a n andq=log a m Thenap=nandaq=m Thereforen/m= ap/aq n/m= ap-q (Law of index subtraction) p-q= log a n/m log a n - log a m= log a n/m

8. The Laws of Logs – Power Law Let p=log a nm thenap=nm ap/m=n Take mth root. So by definition of log p/m=log a n Thus p=m log a n log a nm =m log a n

9. The Laws of Logs – Change of Base Sometime it is useful to change from one base to another Let’s change from base a to base b y=log a x ay= x logb ay= logb x Take logs to base b y logb a= logb x Power law log a x = logb x / logb a

10. Some Special Results log a a = 1 log a 1 = 0 log a x undefined x >= 0

11. Summary y=log a x same as x=ay Definition of log log a n + log a m= log a nm Multiplication Law log a n - log a m= log a n/m Quotient Law log a nm =m log a n Power Law log a x = logb x / logb a Change of Base