Difference between revisions of "Math 22 Logarithmic Functions"
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==Logarithm Function== | ==Logarithm Function== | ||
| − | The logarithm <math>log_a x</math> is defined as | + | The logarithm <math>\log_a x</math> is defined as |
| − | <math>log_a x=b</math> if and only if <math>a^b=x</math> | + | <math>\log_a x=b</math> if and only if <math>a^b=x</math> |
| + | |||
| + | ==Definition of the Natural Logarithmic Function== | ||
| + | The natural logarithmic function, denoted by <math>\ln x</math>, is defined as | ||
| + | <math>\ln x=b</math> if and only if <math>e^b=x</math> | ||
| + | |||
| + | ==Properties of the Natural Logarithmic Function== | ||
| + | Let <math>g(x)=\ln x </math> | ||
| + | 1. The domain of <math>g(x)</math> is <math>(0,\infty)</math> and the range of <math>g(x)</math> is <math>(-\infty,\infty)</math> | ||
| + | 2. The x-intercept of the graph of <math>g(x)</math> is <math>(1,0)</math> | ||
| + | 3. The function <math>g(x)</math> is continuous, increasing, and one-to-one. | ||
| + | 4. <math>\lim_{x\to 0^+} g(x)=-\infty</math> and <math>\lim_{x\to\infty} g(x)=\infty</math> | ||
| + | ==Inverse Properties of Logarithms and Exponents== | ||
| + | 1.<math>\ln e^{\sqrt{2}}</math> | ||
| + | |||
| + | 2.<math>e^{\ln x}=x</math> | ||
| + | |||
| + | 3.<math>\ln{xy}=\ln{x}+\ln{y}</math> | ||
| + | |||
| + | 4.<math>\ln{\frac{x}{y}}=\ln x - \ln y</math> | ||
| + | |||
| + | 5.<math>\ln{x^n}=n\ln x</math> | ||
| + | |||
| + | |||
| + | |||
| + | '''Exercises 1''' Use the properties of logarithms to rewrite the expression as the logarithm of a single quantity | ||
| + | |||
| + | '''a)''' <math>\ln(x-2)-\ln(x+2)</math> | ||
| + | {| class = "mw-collapsible mw-collapsed" style = "text-align:left;" | ||
| + | !Solution: | ||
| + | |- | ||
| + | |<math>\ln(x-2)-\ln(x+2)=\ln \frac{x-2}{x+2}</math> | ||
| + | |} | ||
| + | |||
| + | '''b)''' <math>5\ln (x-6)+\frac{1}{2}\ln(5x+1)</math> | ||
| + | {| class = "mw-collapsible mw-collapsed" style = "text-align:left;" | ||
| + | !Solution: | ||
| + | |- | ||
| + | |<math>5\ln(x-6)+\frac{1}{2}\ln(5x+1)=\ln(x-6)^5+\ln[(5x+1)^{\frac{1}{2}}]=\ln [(x-6)^5\sqrt{5x+1}]</math> | ||
| + | |} | ||
| + | |||
| + | '''c)''' <math>3\ln x+2\ln y -4\ln z</math> | ||
| + | {| class = "mw-collapsible mw-collapsed" style = "text-align:left;" | ||
| + | !Solution: | ||
| + | |- | ||
| + | |<math>\ln x^3 + \ln y^2 -\ln z^4=\ln\frac{x^3y^2}{z^4}</math> | ||
| + | |} | ||
| + | |||
| + | '''d)''' <math>7\ln (5x+4)-\frac{3}{2}\ln (x-9)</math> | ||
| + | {| class = "mw-collapsible mw-collapsed" style = "text-align:left;" | ||
| + | !Solution: | ||
| + | |- | ||
| + | |<math>7\ln (5x+4)-\frac{3}{2}\ln (x-9)=\ln (5x+4)^7-\ln (x-9)^{\frac{3}{2}}=\ln\frac{(5x+4)^7}{(x-9)^{\frac{3}{2}}}</math> | ||
| + | |} | ||
| + | |||
| + | '''Exercises 2''' Solve for x. | ||
| + | |||
| + | '''a)''' <math>\ln(2x)=5</math> | ||
| + | {| class = "mw-collapsible mw-collapsed" style = "text-align:left;" | ||
| + | !Solution: | ||
| + | |- | ||
| + | |<math>\ln(2x)=5</math>, so <math>e^5=2x</math>, hence <math>x=\frac{e^5}{2}</math> | ||
| + | |} | ||
| + | |||
| + | '''b)''' <math>5\ln x=3</math> | ||
| + | {| class = "mw-collapsible mw-collapsed" style = "text-align:left;" | ||
| + | !Solution: | ||
| + | |- | ||
| + | |<math>5\ln x=3</math>, so <math>ln {x^5}=3</math>, so <math>e^3=x^5</math>, hence <math>x=\sqrt[5]{e^3}</math> | ||
| + | |} | ||
| + | |||
| + | |||
[[Math_22| '''Return to Topics Page''']] | [[Math_22| '''Return to Topics Page''']] | ||
'''This page were made by [[Contributors|Tri Phan]]''' | '''This page were made by [[Contributors|Tri Phan]]''' | ||
Latest revision as of 09:44, 11 August 2020
Logarithm Function
The logarithm is defined as
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \log _{a}x=b}
if and only if Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle a^{b}=x}
Definition of the Natural Logarithmic Function
The natural logarithmic function, denoted by , is defined as
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \ln x=b}
if and only if Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle e^{b}=x}
Properties of the Natural Logarithmic Function
Let Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle g(x)=\ln x}
1. The domain of is and the range of is
2. The x-intercept of the graph of is
3. The function is continuous, increasing, and one-to-one.
4. Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \lim _{x\to 0^{+}}g(x)=-\infty }
and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \lim _{x\to \infty }g(x)=\infty }
Inverse Properties of Logarithms and Exponents
1.Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \ln e^{\sqrt {2}}}
2.Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle e^{\ln x}=x}
3.
4.Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \ln {\frac {x}{y}}=\ln x-\ln y}
5.Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \ln {x^{n}}=n\ln x}
Exercises 1 Use the properties of logarithms to rewrite the expression as the logarithm of a single quantity
a) Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \ln(x-2)-\ln(x+2)}
| Solution: |
|---|
| Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \ln(x-2)-\ln(x+2)=\ln {\frac {x-2}{x+2}}} |
b) Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 5\ln(x-6)+{\frac {1}{2}}\ln(5x+1)}
| Solution: |
|---|
![{\displaystyle 5\ln(x-6)+{\frac {1}{2}}\ln(5x+1)=\ln(x-6)^{5}+\ln[(5x+1)^{\frac {1}{2}}]=\ln[(x-6)^{5}{\sqrt {5x+1}}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b548d1076509d504ed47a8fd0003873fa59dff4e)
c) Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 3\ln x+2\ln y-4\ln z}
| Solution: |
|---|
| Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \ln x^{3}+\ln y^{2}-\ln z^{4}=\ln {\frac {x^{3}y^{2}}{z^{4}}}} |
d) Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 7\ln(5x+4)-{\frac {3}{2}}\ln(x-9)}
| Solution: |
|---|
| Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 7\ln(5x+4)-{\frac {3}{2}}\ln(x-9)=\ln(5x+4)^{7}-\ln(x-9)^{\frac {3}{2}}=\ln {\frac {(5x+4)^{7}}{(x-9)^{\frac {3}{2}}}}} |
Exercises 2 Solve for x.
a) Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \ln(2x)=5}
| Solution: |
|---|
| , so Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle e^{5}=2x} , hence Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x={\frac {e^{5}}{2}}} |
b) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 5\ln x=3}
| Solution: |
|---|
| Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 5\ln x=3} , so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ln {x^5}=3} , so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^3=x^5} , hence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=\sqrt[5]{e^3}} |
This page were made by Tri Phan