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.
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.
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
t = = 9.0 years