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Section5.4
Properties of Logarithmic Functions
Section5.4 Properties of Logarithmic Functions - - PowerPoint PPT Presentation
Section5.4 Properties of Logarithmic Functions - - PowerPoint PPT Presentation
Section5.4 Properties of Logarithmic Functions PropertiesofLogarithms Formulas Basic Properties: log a 1 = 0 Formulas Basic Properties: log a 1 = 0 log a a = 1 Formulas Basic Properties: log a 1 = 0 log a a = 1 log a a x = x Formulas Basic
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Formulas
Basic Properties: loga 1 = 0
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Formulas
Basic Properties: loga 1 = 0 loga a = 1
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Formulas
Basic Properties: loga 1 = 0 loga a = 1 loga ax = x
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Formulas
Basic Properties: loga 1 = 0 loga a = 1 loga ax = x aloga x = x
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Formulas
Basic Properties: loga 1 = 0 loga a = 1 loga ax = x aloga x = x The Product Rule: loga(xy) = loga x + loga y
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Formulas
Basic Properties: loga 1 = 0 loga a = 1 loga ax = x aloga x = x The Product Rule: loga(xy) = loga x + loga y The Quotient Rule: loga
- x
y
- = loga x −loga y
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Formulas
Basic Properties: loga 1 = 0 loga a = 1 loga ax = x aloga x = x The Product Rule: loga(xy) = loga x + loga y The Quotient Rule: loga
- x
y
- = loga x −loga y
The Power Rule: loga(xz) = z loga x
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Examples
Calculate the following:
- 1. log4 64
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Examples
Calculate the following:
- 1. log4 64
3
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Examples
Calculate the following:
- 1. log4 64
3
- 2. 10log 51
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Examples
Calculate the following:
- 1. log4 64
3
- 2. 10log 51
51
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Examples
Calculate the following:
- 1. log4 64
3
- 2. 10log 51
51
- 3. log3
√ 27
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Examples
Calculate the following:
- 1. log4 64
3
- 2. 10log 51
51
- 3. log3
√ 27
3 2
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Examples
Calculate the following:
- 1. log4 64
3
- 2. 10log 51
51
- 3. log3
√ 27
3 2
- 4. ln e200
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Examples
Calculate the following:
- 1. log4 64
3
- 2. 10log 51
51
- 3. log3
√ 27
3 2
- 4. ln e200
200
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Examples (continued)
Use the Properties of Logarithms to expand the expression into sums and differences of logarithms:
- 5. log2
2x4 y2z10
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Examples (continued)
Use the Properties of Logarithms to expand the expression into sums and differences of logarithms:
- 5. log2
2x4 y2z10
1 + 4 log2 x − 2 log2 y − 10 log2 z
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Examples (continued)
Use the Properties of Logarithms to expand the expression into sums and differences of logarithms:
- 5. log2
2x4 y2z10
1 + 4 log2 x − 2 log2 y − 10 log2 z
- 6. log4
3
√ x2 − 2x
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Examples (continued)
Use the Properties of Logarithms to expand the expression into sums and differences of logarithms:
- 5. log2
2x4 y2z10
1 + 4 log2 x − 2 log2 y − 10 log2 z
- 6. log4
3
√ x2 − 2x
1 3 log4 x + 1 3 log4(x − 2)
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Examples (continued)
Use the Properties of Logarithms to expand the expression into sums and differences of logarithms:
- 5. log2
2x4 y2z10
1 + 4 log2 x − 2 log2 y − 10 log2 z
- 6. log4
3
√ x2 − 2x
1 3 log4 x + 1 3 log4(x − 2)
- 7. log5
- 3x
y
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Examples (continued)
Use the Properties of Logarithms to expand the expression into sums and differences of logarithms:
- 5. log2
2x4 y2z10
1 + 4 log2 x − 2 log2 y − 10 log2 z
- 6. log4
3
√ x2 − 2x
1 3 log4 x + 1 3 log4(x − 2)
- 7. log5
- 3x
y 1 2 log5 3 + 1 2 log5 x − 1 2 log5 y
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Examples (continued)
Use the Properties of Logarithms to expand the expression into sums and differences of logarithms:
- 5. log2
2x4 y2z10
1 + 4 log2 x − 2 log2 y − 10 log2 z
- 6. log4
3
√ x2 − 2x
1 3 log4 x + 1 3 log4(x − 2)
- 7. log5
- 3x
y 1 2 log5 3 + 1 2 log5 x − 1 2 log5 y
- 8. Given that loga 2 ≈ 0.301 and loga 7 ≈ 0.845, find loga 28.
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Examples (continued)
Use the Properties of Logarithms to expand the expression into sums and differences of logarithms:
- 5. log2
2x4 y2z10
1 + 4 log2 x − 2 log2 y − 10 log2 z
- 6. log4
3
√ x2 − 2x
1 3 log4 x + 1 3 log4(x − 2)
- 7. log5
- 3x
y 1 2 log5 3 + 1 2 log5 x − 1 2 log5 y
- 8. Given that loga 2 ≈ 0.301 and loga 7 ≈ 0.845, find loga 28.
1.447
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Examples (continued)
Use the Properties of Logarithms to combine the expression into a single logarithm:
- 9. −2 ln 3 + 4 ln x − 1
2 ln(x − 1)
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Examples (continued)
Use the Properties of Logarithms to combine the expression into a single logarithm:
- 9. −2 ln 3 + 4 ln x − 1
2 ln(x − 1)
ln
x4 9√x−1
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Examples (continued)
Use the Properties of Logarithms to combine the expression into a single logarithm:
- 9. −2 ln 3 + 4 ln x − 1
2 ln(x − 1)
ln
x4 9√x−1
- 10. log 4900 − 2 log 7
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Examples (continued)
Use the Properties of Logarithms to combine the expression into a single logarithm:
- 9. −2 ln 3 + 4 ln x − 1
2 ln(x − 1)
ln
x4 9√x−1
- 10. log 4900 − 2 log 7
2
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Examples (continued)
Use the Properties of Logarithms to combine the expression into a single logarithm:
- 9. −2 ln 3 + 4 ln x − 1
2 ln(x − 1)
ln
x4 9√x−1
- 10. log 4900 − 2 log 7
2
- 11. 4 log5(x − 4) + log5(x − 1) − 3 log5(x2 − 5x + 4)
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Examples (continued)
Use the Properties of Logarithms to combine the expression into a single logarithm:
- 9. −2 ln 3 + 4 ln x − 1
2 ln(x − 1)
ln
x4 9√x−1
- 10. log 4900 − 2 log 7
2
- 11. 4 log5(x − 4) + log5(x − 1) − 3 log5(x2 − 5x + 4)
log5
x−4 (x−1)2