8.4 Logarithms
By: L. Keali’i Alicea
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 ay = 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 Exp. form
•log216 = 4
•log1010 = 1
•log31 = 0
•log10 .1 = -1
•log2 6 ≈ 2.585
•24 = 16•101 = 10•30 = 1•10-1 = .1•22.585 = 6
Evaluate without a calculator
•log381 =
•Log5125 =
•Log4256 =
•Log2(1/32) =
•3x = 81•5x = 125•4x = 256•2x = (1/32)
4
34
-5
Evaluating logarithms now you try some!
•Log 4 16 = •Log 5 1 =•Log 4 2 =•Log 3 (-1) =• (Think of the graph of y=3x)
20
½ (because 41/2 = 2) undefined
You should learn the following general forms!!!
•Log a 1 = 0 because a0 = 1
•Log a a = 1 because a1 = 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 b x is the inverse of
•f(x) = bx
•f(g(x)) = x and g(f(x)) = x•Exponential and log functions
are inverses and “undo” each other
•So: g(f(x)) = logbbx = x• f(g(x)) = blog
bx = x
•10log2 = •Log39x =•10logx =•Log5125x =
2Log3(32)x =Log332x=2x
x3x
Finding Inverses
• Find the inverse of:
•y = log3x• By definition of logarithm, the inverse
is y=3x • 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) Switch the x &
y
•ex = y + 1 Write in exp form
•ex – 1 = y 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
8.4 A (all)
8.4 B (1-27 odd, 28-29)
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