Logarithmic functions
Definition of a logarithm
log_{b}x = the power to which the base b must be raised to obtain the number x.Example: log_{10}100 = 2
Example: log_{5}125 = 3
Conversion between logarithmic and exponential forms
y = log_{b} x <=> x = b^{y}Example: 2^{3} = 8 is equivalent to log_{2}8 = 3
Example: log_{5}x = 3 is equivalent to x = 5^{3} = 125
Rules of logarithms
log_{b} 1 = 0 |
zero exponent |
log_{b} b = 1 |
unit exponent |
log_{b}(xy) = log_{b}x + log_{b}y |
product rule |
_{ }æ a ö log_{b} ç |
quotient rule |
log_{b}x^{p} = p log_{b}x |
power rule |
Example: Expand ln (sqrt(x)/(x+1)^{2})
ln (sqrt(x)/(x+1)^{2})= ln(x^{1/2}) - ln(x+1)^{2} = 1/2 ln(x) - 2 ln(x+1)
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
Exercises
| Table of Contents | | Previous |