Logarithms

1. The Definition of the Logarithm

 Definition  The function logbx is defined as the inverse function of y = bx

Recall that by definition, if f and g are inverse functions then

f(g(x)) = g(f(x)) = x

Hence we have the following two properties:

 Log Properties  (From the inverse definition) logbbx  =  x blogb(x) = x

Example:

Solve

2x = 128

Solution

Take the log base 2 of both sides:

log22x = log2128

hence

x = log2128

Note that Property 1 allows us to cancel the log and the exponent

Example:

log39 = 2

since

32 = 9

Exercises:

Find

1. log101000

2. log464

3. log51/5

4. log3( )

Simplify

1. 10log10(1/x)

2. log3 27x-1

3. log4(24x-2)

2. Logs and Calculators

Goal:

Find

log317

Note:  The calculator has ln and log

 Definition log x = log10 x  ln x = loge x

 Change of Base Formula logba = lna/lnb = loga/logb

Hence

log317 = ln17/ln3 = 2.5789...

Exercise:

Find

log529

and

log618

3.  Logs and Graphs

Below is the graph of

y = log2

It can be found by reflecting

y  =  2x

across the line

y  =  x

Note:
The domain of the inverse is the range of the function and the range of the inverse is the domain of the function.  Hence,

the domain of log x is (0 ,

and

the range of log x is R

Exercise

Use shifting rules to graph

y  =  log2(x - 3) + 1

and

y  =  -log2x

4. Application

The pH of a liquid describes how acidic or basic the liquid is.  Chemists define the pH by the formula:

pH = -log[H+]

where  [H+] is the concentration of hydrogen ions.

Example

A solution of Hydrochloric acid has

[H+] = 3.2 X 10-4

Find the pH of the solution.

Solution

PH = -log(3.2 X 10-4)  =  3.5

Exercise

Suppose that the pH of a shampoo is 7.3.  Find the concentration of hydrogen ions.