Math 112  19:  Logs and Exponential Functions

 

1.  Review of Logs.  Logab = c, what does it mean and why do we care?

One primary objective of math is to solve equations.

Examples:      3x + 5 = 4

                       x2 - 2x - 15 = 0

                       sin x = .5

                       3x = 5

The last two equations require a new method:  Writing equations in inverse form.

 

i.e. to solve sin x = .5, we write "x is the angle whose sin is .5" as sin-1.5 = x, and

to solve 3x = 5 we write "the power to which 3 is raised to get 5 is x" as log35 = x.

 

The Natural log function and the number e.

 

e = limit (1 + 1/n)n  = 2.718281828...   it is a non-repeating decimal, hence irrational.

      n°

ex = limit (1 + x/n)n

       n°

logex is denoted ln x and is called the natural log function.  logex ln x

 

Drill on saying and doing.  Solve for x:

 

log101000 = x

log82 = x

log40.5 = x

log5(1/125) = x

log2_2 = x

log101 = x

logb 1 = x

logx16 = 4

log5 x = 3

log.5x = 2

log125 x = 2/3

logx 27 = 3/2

log10 x = 0

log5 0 = x

log3 (-9) = x

Laws of Logs:

 

1.  log ab = log a + log b

2.  log a/b = log a - log b

3.  log ab = b log a

Change of base formula

 4.  

Solve for x:

 

1.  log x = 3log 2 - log 4

2.  log3 x = .5log35 + log34 -2log33

3.  log2 x = log26 - .5log2 9

4.  ln x = 3 ln2 - 2 ln 3 - ln 6

5.  ln x2 - ln 2x = 3 ln 3 - ln6