Logarithms

I.  Homework

II.  The Definition of the Logarithm

Definition:  The function log_bx is defined as the inverse function of y = b^x   

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:

1)  log_bb^x  =  x

2)  b^logb(x) = x 

Example:  Solve 2^x = 128

Solution  Take the log base 2 of both sides:

log₂2^x = 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

A)  log₁₀1000

B)  log₄64

C)  log₅1/5

D)  log₃(sqrt(3))

Simplify

A)  10^log10(1/x) 

B) log₃ 27^x-1

C)  log₄(2^4x-2)    

III.  Logs and Calculators

Goal:  Find log₃17

Note:  The calculator has ln and log

Definition:

1)  log x = log₁₀ x 

2)  ln x = log_e x

Change of Base Formula:

log_ba = lna/lnb = loga/logb

Hence   log₃17 = ln17/ln3 = 2.5789...

Exercise:  Find log₅29 and log₆18

IV.  Logs and Graphs

We will graph y = log₂ x using the reflection property of inverses.  

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,infinity) and the range of log x is R     

We will use shifting rules to graph

log₂(x - 3) + 1 and -log₂x  

V.  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.

1)  a solution of Hydrochloric acid has

[H⁺] = 3.2 X 10^-4

Find the pH of the solution.

2) Suppose that the pH of a shampoo is 7.3

find the concentration of hydrogen ions.