1.5 - LOGARITHMS
I. Inverse Functions - The inverse of a function f , denoted
f 
1
,
reverses the effect of f.
Not every function has an inverse. Only functions that are one-to-one have inverses, i.e. passes the horizontal
line test.
The range of f becomes the domain of
f  1 and domain of
f becomes the range of
f 1 .
II. LOGARITHMIC FUCNTION
Graph of
f(x)= logb x , b > 0, b  1
Properties:
Continuous on
( 0,  )
Passes through (1, 0 )
Domain:
( 0,  )
Range:

b > 1 - increasing
0 < b < 1 - decreasing
III. Conversions
loga x = c or ac = x
So, LOGARITHMS ARE EXPONENTS!!
(defined for x > 0; recall graph)
A. 10 is a special base (called the common log) and is written
logx = c or 10c = x
This says that the logarithm to base 10 of x is the power of 10 you need to get x. Examples of base 10:
Richter scale, pH
B. The most frequently used base is base e or the natural logarithm.
loge x = ln x = c or ec = x
(ln x is the power of e needed to get x)
III. PROPERTIES OF LOGARITHMS
1.
logb (xy)= logb x + logb y
2.
 x
logb   = logb x  logb y
 y
3.
logb x p = plogb x
4.
logbb x = x
log x
5. b b = x
6.
loga x =
IV. Change of Base Formula:
Function Hierarchy:
ex
x
grows the fastest
p
ln x
Examples:
(p + )
grows slowest
logb1= 0 becauseb0 = 1
logbb = 1 becauseb1 = b
ln x
ln a
and
e xln b = b x
1.5 HW # 1 – 43 (odd), 48, 49
Ch. 1 review p.93 # 11,15,17,19, 37 - 47 (odd), 48