Download Logarithms

XV. TRANSCENDENTAL FUNCTIONS- EXPONENTIAL AND LOGARITHMIC
The exponential function f with base a is denoted by f  x   a x where a  0, a  1, and x is any
real number.
The exponential function f  x   a x is different from all other functions you have studied because the
variable x is an exponent. The domain, like those of polynomial functions, is the set of all real
numbers. A description of the graph of f  x   a x is included in the LIBRARY OF FUNCTIONS
SUMMARY at the end of this packet.
For many applications that we will study next year, (continuously compounded interest, population
growth, radioactive decay), the best choice for a base in the exponential function f  x   a x , is the
irrational number e  2.71828...
e called the natural base and the function f  x   e x is the natural exponential function.
The number e can be approximated by the expression: 1  1x  for large values of x.
x
Because the exponential function f  x   a x is a one-to-one function, it must have an inverse function.
This inverse function is called the logarithmic function with base a.
y  log a x if and only if x  a y
The function
f  x   loga x is called the logarithmic function with base a.
When evaluating logarithms, remember that a logarithm is an exponent. This means that log a x is the
exponent to which a must be raised in order to obtain x.
Example:
Example:
log5 25  2 because 5 must be raised to the second power to get 25.
log3 13   1 because 3 must be raised to the negative one power to get
Properties of Logarithms
log a 1  0 because a 0  1
u
log a    log a u  log a v
v
log a a  1 because a  a
1
log a a x  x and a loga x  x
Properties)
(Inverse
If log a x  log a y, then x  y (one-to-one
property)
loga  uv   loga u  loga v
log a u n  n log a u
1
3
Special Logarithms
Common logarithm- logarithmic function with base 10. On most calculators, this
function is denoted by LOG
Example: log 100 = 2 because 102  100
Example: log.1   1 because 101  101  .1
Natural logarithm- logarithmic function with base e. The natural logarithmic
function f  x   loge x  ln x . On most calculators, the natural logarithm is denoted by
LN
Example: ln e2  2 because log e e2  2
Example: ln 2  .69315 because e.69315  2.71828.69315  2
Change of Base Formula
To compute any logarithm, you may change the base of the logarithm to one of the
special logs (common log or natual log).
log a x 
log x
ln x

log a
ln a
log 30 1.47712

 2.4534
log 4
0.60206
ln 30
3.40120
log 4 30 

 2.4535
ln 4
1.38629
Example: log 4 30 
OR
*NOTE- The decimal approximations are almost identical using either change of base
formula!