8.4 Logarithms
p. 486
Evaluating Log Expressions
• We know 22 = 4 and 23 = 8
• But for what value of y does 2y = 6?
• Because 22<6<23 you would expect
the answer to be between 2 & 3.
• To answer this question exactly,
mathematicians defined logarithms.
Definition of Logarithm to base a
• Let a & x be positive numbers & a ≠ 1.
• The logarithm of x with base a is
denoted by logax and is defined:
• logax = y iff
y
a
=x
• This expression is read “log base a of x”
• The function f(x) = logax is the
logarithmic function with base a.
• The definition tells you that the
equations logax = y and ay = x are
equivilant.
• Rewriting forms:
• To evaluate log3 9 = x ask yourself…
• “Self… 3 to what power is 9?”
• 32 = 9 so……
log39 = 2
Log form
• log216 = 4
• log1010 = 1
• log31 = 0
• log10 .1 = -1
• log2 6 ≈ 2.585
Exp. form
• 24 = 16
1
• 10 = 10
0
•3 = 1
• 10-1 = .1
• 22.585 = 6
Evaluate without a calculator
• log381 = 4
• Log5125 = 3
• Log4256 = 4
• Log2(1/32) =
-5
x
•3
= 81
x
• 5 = 125
• 4x = 256
x
• 2 = (1/32)
Evaluating logarithms now you try
some!
• Log 4 16 = 2
• Log 5 1 = 0
• Log 4 2 = ½ (because 41/2 = 2)
• Log 3 (-1) = undefined
• (Think of the graph of y=3x)
You should learn the following
general forms!!!
0
a
• Log a 1 = 0 because = 1
1
• Log a a = 1 because a = a
• Log a ax = x because ax = ax
Natural logarithms
•log e x = ln x
• ln means log base e
Common logarithms
•log 10 x = log x
• Understood base 10 if
nothing is there.
Common logs and natural logs with
a calculator
log10 button
ln button
• g(x) = log
• f(x) = bx
b
x is the inverse of
• f(g(x)) = x and g(f(x)) = x
• Exponential and log functions
are inverses and “undo” each
other
• So: g(f(x)) = logb
•
x
b
=x
log
x
f(g(x)) = b b = x
• 10log2 = 2
x
• Log39 = Log3(32)x =Log332x=2x
logx
• 10
= x
x
• Log5125 = 3x
Finding Inverses
• Find the inverse of:
• y = log3x
• By definition of logarithm, the inverse is
x
y=3
• OR write it in exponential form and
switch the x & y!
3y = x
3x = y
Finding Inverses cont.
• Find the inverse of :
• Y = ln (x +1)
• X = ln (y + 1)
• ex = y + 1
• ex – 1 = y
Switch the x & y
Write in exp form
solve for y
Assignment
Graphs of logs
• y = logb(x-h)+k
• Has vertical asymptote x=h
• The domain is x>h, the range is
all reals
• If b>1, the graph moves up to
the right
• If 0<b<1, the graph moves down
to the right
Graph y = log1/3x-1
• Plot (1/3,0) & (3,-2)
• Vert line x=0 is asy.
• Connect the dots
X=0
Graph y =log5(x+2)
• Plot easy points
(-1,0) & (3,1)
• Label the
asymptote x=-2
• Connect the
dots using the
asymptote.
X=-2
Assignment