Difference between revisions of "Math 22 Logarithmic Functions"
Jump to navigation
Jump to search
| Line 1: | Line 1: | ||
==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== | ==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== | ==Properties of the Natural Logarithmic Function== | ||
| − | Let <math>g(x)=ln x </math> | + | 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> | 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> | 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. | 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> | 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> | ||
| + | |||
Revision as of 07:58, 11 August 2020
Logarithm Function
The logarithm is defined as if and only if
Definition of the Natural Logarithmic Function
The natural logarithmic function, denoted by , is defined as if and only if
Properties of the Natural Logarithmic Function
Let 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. and
Inverse Properties of Logarithms and Exponents
1. 2.
This page were made by Tri Phan