Math 22 Logarithmic Functions - Revision history

Tphan046: /* Inverse Properties of Logarithms and Exponents */

2020-08-11T16:44:45Z

Inverse Properties of Logarithms and Exponents

2020-08-11T16:44:19Z

Inverse Properties of Logarithms and Exponents

2020-08-11T16:36:42Z

Inverse Properties of Logarithms and Exponents

2020-08-11T16:34:35Z

Inverse Properties of Logarithms and Exponents

2020-08-11T16:31:18Z

Inverse Properties of Logarithms and Exponents

2020-08-11T16:27:22Z

2020-08-11T16:01:11Z

Inverse Properties of Logarithms and Exponents

2020-08-11T15:58:33Z

Inverse Properties of Logarithms and Exponents

2020-08-11T15:58:18Z

2020-08-11T15:54:15Z

← Older revision		Revision as of 16:44, 11 August 2020
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	5.<math>\ln{x^n}=n\ln x</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		'''Exercises 1''' Use the properties of logarithms to rewrite the expression as the logarithm of a single quantity

@@ Line 24: / Line 24: @@
 .<math>\ln{x^n}=n\ln x</math>
-'''Exercises''' Use the properties of logarithms to rewrite the expression as the logarithm of a single quantity
+'''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>
@@ Line 53: / Line 53: @@
 |<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>
 |}
 [[Math_22| '''Return to Topics Page''']]
 '''This page were made by [[Contributors|Tri Phan]]'''

@@ Line 51: / Line 51: @@
 !Solution: &nbsp;
 |-
-|<math>f'(x)=6xe^{3x^2}</math>
+|<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>
 |}

@@ Line 37: / Line 37: @@
 !Solution: &nbsp;
 |-
-|<math>5\ln{x-6}+\frac{1}{2}\ln(5x+1)=\ln(x-6)^5+\ln[(5x+1)^{\frac{1}{2}}]</math>
+|<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>
 |}
@@ Line 44: / Line 44: @@
 !Solution: &nbsp;
 |-
-|<math>f'(x)=6xe^{3x^2}</math>
+|<math>\ln x^3 + \ln y^2 -\ln z^4=\ln\frac{x^3y^2}{z^4}</math>
 |}

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 !Solution: &nbsp;
 |-
-|<math>f'(x)=2e^{2x}</math>
+|<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>
+'''b)''' <math>5\ln (x-6)+\frac{1}{2}\ln(5x+1)</math>
 {| class = "mw-collapsible mw-collapsed" style = "text-align:left;"
 !Solution: &nbsp;
 |-
-|<math>f'(x)=6xe^{3x^2}</math>
+|<math>5\ln{x-6}+\frac{1}{2}\ln(5x+1)=\ln(x-6)^5+\ln[(5x+1)^{\frac{1}{2}}]</math>
 |}

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 ==Inverse Properties of Logarithms and Exponents==
 .<math>\ln e^{\sqrt{2}}</math>
 .<math>e^{\ln x}=x</math>

@@ Line 1: / Line 1: @@
 ==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
+   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>
+   <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>
+   Let <math>g(x)=\ln x </math>
 . 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>
 . The x-intercept of the graph of <math>g(x)</math> is <math>(1,0)</math>
 . The function <math>g(x)</math> is continuous, increasing, and one-to-one.
 . <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==
+.<math>\ln e^{\sqrt{2}}</math>