Example: Solve log_{3}(x + 1) + log_{3}(2x - 3) = 1 for the unknown in the logarithm

Combine into a single logarithm using the product rule
log_{3}[(x + 1)(2x - 3)] = 1
Convert to exponential form
(x + 1)(2x - 3) = 3^{1}
2 x^{2} - x - 3 = 3
2 x^{2} - 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 10^{x} = 500 for the unknown in the exponent.

Clearly 10^{x} = 100 has solution x = 2 and 10^{x} = 1000 has solution x = 3.
To find the solution of 10^{x} = 500 is more difficult. We can approximate an answer by
taking an inital guess of x = 2.5 and then refining the answer. Since 10^{2.5} = 316.2 we
increase to x = 2.7 to obtain 10^{2.5} = 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 10^{x} = 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.

log_{10}x = log x = common logarithm
log_{e}x = 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.

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 1.08^{t}
Use the power rule of logarithms
ln 2 = t ln 1.08
Solve for t
ln 2
t = = 9.0 years
ln 1.08