Transcript Logarithms

Logarithms
The Inverse
• Logarithms are the “opposite” of exponentials (like
subtraction and addition or division and multiplication).
Logs “undo” exponentials. Technically speaking, logs
are the inverses of exponentials.
• The logarithm of a number to a given base is the power
or exponent to which the base must be raised in order to
produce the number. For example, the logarithm of 1000
to the base of 10 is 3. We write it as
The graph of
• and it's inverse.
• The inverse
is the logarithm.
Asymptotes?
The Relationship
y=
x
b
is equivalent logb(y) = x
to
• Note that b is the base and x is the exponent.
Convert 6³ = 216 to the equivalent
logarithmic expression.
y = bx
is equivalent to
• log6(216) = 3
• 6³ = 216
• Worksheet #1-6
logb(y) = x
Convert log4(1024) = 5 to the
equivalent exponential expression.
What do we need to use?
Convert log4(1024) = 5 to the
equivalent exponential expression.
y = bx
is equivalent to
logb(y) = x
Convert log4(1024) = 5 to the
equivalent exponential expression.
y = bx
is equivalent to
• 45 = 1024
• You do…worksheet #7-12
logb(y) = x
Simplify log2(8)
• Set the equation equal to x: log2(8) = x
• We know that
y = bx
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•
•
•
•
is equivalent to
logb(y) = x
then 2 x = 8
Solve for x:
x=3
Therefore, log2(8) = 3
You do … worksheet #13-18
Simplify
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•
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•
A: log7(7)
B: log3(3)
C: log6(6)
D: log12(12)
• What does this tell us about logb(b)?
logb(b) = 1
Simplify
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A. log7(1)
B. log4(1)
C. log3(1)
D. log (1)
• What does this tell us about logb(1) ?
logb(1) = 0
Check these out on your
calculator!
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log(1)
log(10)
log(100)
log(1000)
log(2)
log(20)
log(200)
log(2000)
log(3)
log(30)
log(300)
log(3000)
Zeroes
y = bx
• Simplify:
• F: log4(0)
• G: log3(0)
is equivalent to
logb(y) = x
Zeroes
y = bx
is equivalent to
logb(y) = x
• Simplify:
• F: log4(0) No solution
• G: log3(0) No solution
• What does this tell us about logb(0)?
• logb(0) is undefined for any base b,
not just for b = 4 or b = 3.
Simplify logb
y = bx
is equivalent to
3
(b )
logb(y) = x
• Because "logb(b3) = x" means "b x = b3",
then x = 3, so logb(b3) = 3.
• What does this tell us about logb(bn)?
logb(bn) = n for any base b.
Summary
"logb(y) = x" means the same thing as "b x = y".
Logarithms "undo" exponents (powers).
logb(b) = 1, for any base b, because b1 = b.
logb(1) = 0, for any base b, because b0 = 1.
logb(a) is undefined if a is negative.
logb(0) is undefined for any base b.
logb(bn) = n, for any base b.
Remember e?
• A logarithm with a base of e is called a
natural logarithm and is denoted by ln.
ln 42 = 3.74
ln e = ?
ln 0 = ?
ln e³ = ?
ln –e = ?
Graphing
• Graph log3(x + 1) - 3
Graphing
• Graph log2(x - 4) + 2