Definition of a logarithmlogbx = 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 formsy = 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
logb b = 1
logb(xy) = logbx + logby
æ a ö logb ç
logbxp = p logbx
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 functionsBase 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.
Business applicationsLogarithms 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