Logarithmic functions

Definition of a logarithm

logbx = the power to which the base b must be raised to obtain the number x.

Example:     log10100 = 2

Example:     log5125 = 3

Conversion between logarithmic and exponential forms

y = logb x <=> x = by

Example:     23 = 8 is equivalent to log28 = 3

Example:     log5x = 3 is equivalent to x = 53 = 125

Rules of logarithms 


 logb 1 = 0
 zero exponent 

 logb b = 1
 unit exponent 

 logb(xy) = logbx + logby
 product rule 

       æ a ö   
 logb  ç   ÷ = logbx - logby  
       è b ø
quotient rule 

 logbxp = p logbx
 power rule

Example:     Expand ln (sqrt(x)/(x+1)2)

ln (sqrt(x)/(x+1)2)= ln(x1/2) - ln(x+1)2 = 1/2 ln(x) - 2 ln(x+1)


Example:     Solve log3(x + 1) + log3(2x - 3) = 1 for the unknown in the logarithm 
Combine into a single logarithm using the product rule
log3[(x + 1)(2x - 3)] = 1
Convert to exponential form
(x + 1)(2x - 3) = 31
2 x2 - x - 3 = 3
2 x2 - x - 6 = 0
(x - 2)(2x + 3) = 0
x = 2, -3/2
Substitute the answers into the original equation.
Since we cannot take the logarithm of a negative number 
we choose x = 2.
Example:     Solve 10x = 500 for the unknown in the exponent.

Clearly 10x = 100 has solution x = 2 and 10x = 1000 has solution x = 3. To find the solution of 10x = 500 is more difficult. We can approximate an answer by taking an inital guess of x = 2.5 and then refining the answer. Since 102.5 = 316.2 we increase to x = 2.7 to obtain 102.5 = 501.19. Since this is too large we should decrease x slightly. Continuing in this manner we may obtain an approximate answer. We can use logarithms to solve the equation in a more systemic manner. Take the common logarithm of both sides to obtain log 10x = log 500. Apply the power rule x log 10 = log 500 and use log 10 = 1 to obtain x = log 500. Using a scientific calculator yields x = 2.69897.

Logarithmic functions

Base e and base 10 are the most important logarithms and most calculators have these two logarithms.

log10x = log x = common logarithm
logex = ln x = natural logarithm

The graphs of logarithmic functions are defined only for x > 0.
The function grows rapidly on 0 < x < 1 and slowly for x > 1.
Logarithmic function

Business applications

Logarithms can be used to compute the doubling or tripling time of an investment.

Example:     Suppose $1000 is invested at an interest rate of 8% compounded
annually. Compute the time required for the investment to double in value. 
 
The amount of money in the bank after t years is
A = 1000 (1.08)t
Set the final amount to $2,000 (doubling time).
2000 = 1000 (1.08)t OR
2 = (1.08)t
Take a logarithm on both sides
ln 2 = ln 1.08t
Use the power rule of logarithms
ln 2 = t ln 1.08
Solve for t
      ln 2
t =          = 9.0 years
     ln 1.08

Exercises

(1) log101000 = 
2 
3 
10 
100

 (2) Use the rules of logarithms to expand log (x*sqrt(x2 + 1)). 
log(x) + log(sqrt(x2 + 1)) 
log x - log(sqrt(x2 + 1)) 
log x - 1/2 log x2 - 1/2 log 1 
log x - 1/2 log (x2 + 1)  

(3) Solve log2(2x + 6) = 4 for the unknown in the logarithm. 
x = 1 
x = 5 
x = 3 
x = 4 

 (4) Solve 2 3x = 36 for the unknown in the exponent. 
x = log 18/log 3 
x = log 6 
x = log 36/log 6 
x = 2  

(5) The graph of f(x) = log x could be Logarithmic function graph
    a)
    b) 
    c)
    d) 


(5) Suppose $1,000 is placed in an account earning 6% interest
    compounded annually.  The amount of money in the account as
    as function of time is given by A = 1,000 (1.06)t.  Find the 
    time required for the value of this investment to double.
    11.9 years
    1.2 years
    1.9 years
    89.3 years

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