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Evaluating logarithms

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Evaluating logarithms A logarithm is an exponent if there is no base listed, the common log base is 10. Evaluate, Using a Calculator: When you can t rewrite using ... – PowerPoint PPT presentation

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Title: Evaluating logarithms


1
Evaluating logarithms
A logarithm is an exponentif there is no base
listed, the common log base is
10. Evaluate, Using a Calculator
2
Evaluating logarithms
A logarithm is an exponentif there is no base
listed, the common log base is 10. Evaluate
3
Using the calculator-evaluate to the nearest
hundredth
4
Using the calculator
5
The natural logarithm ln
The natural log is base e
6
The natural logarithm ln
The natural log is base e
7
Logarithmic exponential
Log base value exponent Log 2 8 3
23 8 Solve for x Log 6 x 3
8
Logarithmic exponential
Log base value exponent Log 2 8 3
23 8 Solve for x Log 6 x 3 63
x x 216
9
Change of base law
When not a common logarithm use the change of
base Law
10
Change of base law
When not a common logarithm use the change of
base Law
11
Numerical example-evaluate
12
Laws of Logarithms
13
Example from previous slide..
But this is also true
14
Example
15
Examples expand using log laws
16
Examples
17
Single logarithms
18
Single logarithmsdivision, then multiplication
19
Express as a single logarithm
20
Express as a single logarithm
21
Expansion with substitution
Given
Find
First expand then substitute
22
Expansion with substitution
First expand then substitute
23
With numbers.
Rewrite the numbers in terms of factors of 4 and
5 only
24
With numbers.
Rewrite the numbers in terms of factors of 4 and
5 And use and find
25
solution
Try page 335 46, 50,52
26
Solving log equations the domain of y log x
is that xgt0!
Do Now1. solve the exponential equation
2. Solve the logarithmic equation
27
Solving log equations when the base is a variable
Recall to raise both change to an sides
to the reciprocal power exponential equation
28
Solving log equations.(combine the last 2
concepts)
  1. Change to an exponential equation
  2. Raise to the reciprocal power.
  3. Check all answers!! (Solutions may be extraneous)

29
Solving log equations.
  1. Change to an exponential equation
  2. Raise to the reciprocal power.

30
More examples in ex. 1, note that the domain is
x-2gt0 soxgt2
31
Solutions
1.
2.
32
Express as a single log first!
33
Express as a single log first!
-2 is extraneous!
34
When you cant rewrite using the same base, you
can solve by taking a log of both sides
2x 7 log 2x log 7 x log 2 log 7 x
2.807
35
Example 4x 15
36
Example 4x 15
log 4x log 15 x log 4 log15 x log 15/log
4 1.95
37
Using logarithms to solve exponentials
Here we should ln both sides.
38
Using logarithms to solve exponentials
39
Solving with logs-isolate first
40
Solving with logs-isolate first
41
Isolate the base term first!
102x 4 21
42
Isolate the base term first!
102x 4 21 102x 17 log 102xlog 17
43
Isolate the base term first!
102x 4 21 102x 17 log 102xlog 17 2xlog 10
log 17
Use ( )!
44
Graphs of exponentials
  • Growth and decay
  • growth decay

45
Compound Formula
  • Interest rate formula

46
Compound Formula
  • How long will it take 200 to become 250 at 5
    interest rate, compounded quarterly

47
Compound Formula
  • How long will it take 200 to become 250 at 5
    interest rate, compounded quarterly

48
solution
  • Log both sides and round to the nearest year

49
CONTINUOUS growth Ex population grows
continuously at a rate of 2 in Allentown. If
Allentown has 10,000 people today, how many years
will it take To have about 11,000 to the nearest
tenth of a year?
50
CONTINUOUS growth
51
Solving Log Equations
  • To solve use the property for logs w/ the same
    base
  • If logbx logby, then x y

52
log3(5x-1) log3(x7)
  • Solve by decompressing

53
log3(5x-1) log3(x7)
  • 5x 1 x 7
  • 5x x 8
  • 4x 8
  • x 2 and check
  • log3(52-1) log3(27)
  • log39 log39

54
Example
  • Solve

55
Example
  • Solve

56
log5x log(x1)log100
  • Decompress

57
log5x log(x1)log100
  • (5x)(x1) 100 (product property)
  • (5x2 5x) 100
  • 5x2 5x-100 0
  • x2 x - 20 0 (subtract 100 and
    divide by 5)
  • (x5)(x-4) 0 x-5, x4
  • 4x is the only solution

58
another
  • Solve

59
another
  • Solve

60
One More!log2x log2(x-7) 3
  • Solve and check


61
One More!log2x log2(x-7) 3
  • log2x(x-7) 3
  • log2 (x2- 7x) 3
  • 2log2(x -7x) 23
  • x2 7x 8
  • x2 7x 8 0
  • (x-8)(x1)0
  • x8 x -1

2
62
Graphs of exponentials
  • Growth and decay
  • growth decay

63
Inverse functions
  • Inverse functions are a reflection in yx

Y2x
Ylog2x
Yx
Domain of y2x is all reals Domain of y log2x
is
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