CH 4.4 Evaluate Logarithms and Graph Logarithmic Functions ...
CH 4.4 Evaluate Logarithms and Graph Logarithmic Functions.notebook December 14, 2012 Evaluate Logarithms and Graph Logarithmic Functions
4.4 Evaluate Logarithms and Graph Logarithmic Functions • Definition of a logarithm
Evaluate Logarithms and Graph Logarithmic Functions
Definition: y is the logarithm of x to the base 10 or log of x to the base 10, written
• Common logs • Natural logs • Inverse properties • Graphs of common and natural logs
Dec 92:10 PM
Evaluate Logarithms and Graph Logarithmic Functions
Logarithms to the base 10 are called common logarithms or common logs. They have their own key on you calculator.
Dec 92:10 PM
Evaluate Logarithms and Graph Logarithmic Functions
iff
Dec 92:10 PM
Evaluate Logarithms and Graph Logarithmic Functions
From the graphing exercise, we have seen that y=10 x and the common log are inverses. Look at the LOG key on your calculator, and above it is 10 x.
Dec 92:10 PM
Evaluate Logarithms and Graph Logarithmic Functions
Properties of y=10x and y=log x 1. The domain of the exponential function is the set of real numbers; its range is the set of positive real numbers. Consequently, the domain of the logarithmic function is the set of positive real numbers; its range is the set of real numbers.
2. The graph of y=10x never touches the xaxis; the xaxis is the asymptote of the graph. Consequently, the graph of y=log x never touches the yaxis; the yaxis is the asymptote of the graph.
3.
The yintercept of y=10x is 1. Consequently, the xintercept of y=log x is 1.
Because they are inverses, each property of the exponential function y=10x corresponds to a property of its inverse, the common logarithm function y=log x.
Dec 92:10 PM
Dec 92:10 PM
1
CH 4.4 Evaluate Logarithms and Graph Logarithmic Functions.notebook December 14, 2012
Evaluate Logarithms and Graph Logarithmic Functions
Dec 92:10 PM
Evaluate Logarithms and Graph Logarithmic Functions
Dec 92:10 PM
Evaluate Logarithms and Graph Logarithmic Functions
Evaluate Logarithms and Graph Logarithmic Functions
Dec 92:10 PM
Evaluate Logarithms and Graph Logarithmic Functions
Dec 92:10 PM
Evaluate Logarithms and Graph Logarithmic Functions
We can now look at logarithms with any base.
iff
Dec 92:10 PM
Dec 92:10 PM
2
CH 4.4 Evaluate Logarithms and Graph Logarithmic Functions.notebook December 14, 2012
Evaluate Logarithms and Graph Logarithmic Functions
Dec 92:10 PM
Evaluate Logarithms and Graph Logarithmic Functions
Dec 92:10 PM
Evaluate Logarithms and Graph Logarithmic Functions
Evaluate Logarithms and Graph Logarithmic Functions
Dec 92:10 PM
Evaluate Logarithms and Graph Logarithmic Functions
Dec 92:10 PM
Evaluate Logarithms and Graph Logarithmic Functions
There are also logarithms with base e. These are called natural logarithms.
iff
Dec 92:10 PM
Dec 92:10 PM
3
CH 4.4 Evaluate Logarithms and Graph Logarithmic Functions.notebook December 14, 2012
Evaluate Logarithms and Graph Logarithmic Functions
Dec 92:10 PM
Evaluate Logarithms and Graph Logarithmic Functions
Evaluate Logarithms and Graph Logarithmic Functions
Dec 92:10 PM
Evaluate Logarithms and Graph Logarithmic Functions
There are some special characteristics of logarithms. When you take the log of a number that is that same as the base of the log, it is always equal to one. The natural log example is especially useful in problem solving.
Dec 92:10 PM
Evaluate Logarithms and Graph Logarithmic Functions
Over time, the average diameter of a certain artery decreases and can be modeled by the equation:
Dec 92:10 PM
Evaluate Logarithms and Graph Logarithmic Functions
What is the average diameter of the certain artery of a 65 year old person.
where a is age in years and d is diameter in micrometers.
Dec 92:10 PM
Dec 92:10 PM
4
CH 4.4 Evaluate Logarithms and Graph Logarithmic Functions.notebook December 14, 2012
Evaluate Logarithms and Graph Logarithmic Functions
Evaluate Logarithms and Graph Logarithmic Functions
p252 Example #4
The wind speed s ( in miles per hour) near the center of a tornado can be modeled by
where d is the distance (in miles) that the tornado travels. In 1925, a tornado traveled 220 miles through three states. Estimate the wind speed near the center of the tornado.
Dec 92:10 PM
Evaluate Logarithms and Graph Logarithmic Functions
Dec 92:10 PM
Evaluate Logarithms and Graph Logarithmic Functions
4.4 Evaluate Logarithms and Graph Logarithmic Functions • Definition of a logarithm • Common logs • Natural logs • Inverse properties • Graphs of common and natural logs
Dec 92:10 PM
Dec 92:10 PM
5
4.4 Evaluate Logarithms and Graph Logarithmic Functions • Definition of a logarithm
Evaluate Logarithms and Graph Logarithmic Functions
Definition: y is the logarithm of x to the base 10 or log of x to the base 10, written
• Common logs • Natural logs • Inverse properties • Graphs of common and natural logs
Dec 92:10 PM
Evaluate Logarithms and Graph Logarithmic Functions
Logarithms to the base 10 are called common logarithms or common logs. They have their own key on you calculator.
Dec 92:10 PM
Evaluate Logarithms and Graph Logarithmic Functions
iff
Dec 92:10 PM
Evaluate Logarithms and Graph Logarithmic Functions
From the graphing exercise, we have seen that y=10 x and the common log are inverses. Look at the LOG key on your calculator, and above it is 10 x.
Dec 92:10 PM
Evaluate Logarithms and Graph Logarithmic Functions
Properties of y=10x and y=log x 1. The domain of the exponential function is the set of real numbers; its range is the set of positive real numbers. Consequently, the domain of the logarithmic function is the set of positive real numbers; its range is the set of real numbers.
2. The graph of y=10x never touches the xaxis; the xaxis is the asymptote of the graph. Consequently, the graph of y=log x never touches the yaxis; the yaxis is the asymptote of the graph.
3.
The yintercept of y=10x is 1. Consequently, the xintercept of y=log x is 1.
Because they are inverses, each property of the exponential function y=10x corresponds to a property of its inverse, the common logarithm function y=log x.
Dec 92:10 PM
Dec 92:10 PM
1
CH 4.4 Evaluate Logarithms and Graph Logarithmic Functions.notebook December 14, 2012
Evaluate Logarithms and Graph Logarithmic Functions
Dec 92:10 PM
Evaluate Logarithms and Graph Logarithmic Functions
Dec 92:10 PM
Evaluate Logarithms and Graph Logarithmic Functions
Evaluate Logarithms and Graph Logarithmic Functions
Dec 92:10 PM
Evaluate Logarithms and Graph Logarithmic Functions
Dec 92:10 PM
Evaluate Logarithms and Graph Logarithmic Functions
We can now look at logarithms with any base.
iff
Dec 92:10 PM
Dec 92:10 PM
2
CH 4.4 Evaluate Logarithms and Graph Logarithmic Functions.notebook December 14, 2012
Evaluate Logarithms and Graph Logarithmic Functions
Dec 92:10 PM
Evaluate Logarithms and Graph Logarithmic Functions
Dec 92:10 PM
Evaluate Logarithms and Graph Logarithmic Functions
Evaluate Logarithms and Graph Logarithmic Functions
Dec 92:10 PM
Evaluate Logarithms and Graph Logarithmic Functions
Dec 92:10 PM
Evaluate Logarithms and Graph Logarithmic Functions
There are also logarithms with base e. These are called natural logarithms.
iff
Dec 92:10 PM
Dec 92:10 PM
3
CH 4.4 Evaluate Logarithms and Graph Logarithmic Functions.notebook December 14, 2012
Evaluate Logarithms and Graph Logarithmic Functions
Dec 92:10 PM
Evaluate Logarithms and Graph Logarithmic Functions
Evaluate Logarithms and Graph Logarithmic Functions
Dec 92:10 PM
Evaluate Logarithms and Graph Logarithmic Functions
There are some special characteristics of logarithms. When you take the log of a number that is that same as the base of the log, it is always equal to one. The natural log example is especially useful in problem solving.
Dec 92:10 PM
Evaluate Logarithms and Graph Logarithmic Functions
Over time, the average diameter of a certain artery decreases and can be modeled by the equation:
Dec 92:10 PM
Evaluate Logarithms and Graph Logarithmic Functions
What is the average diameter of the certain artery of a 65 year old person.
where a is age in years and d is diameter in micrometers.
Dec 92:10 PM
Dec 92:10 PM
4
CH 4.4 Evaluate Logarithms and Graph Logarithmic Functions.notebook December 14, 2012
Evaluate Logarithms and Graph Logarithmic Functions
Evaluate Logarithms and Graph Logarithmic Functions
p252 Example #4
The wind speed s ( in miles per hour) near the center of a tornado can be modeled by
where d is the distance (in miles) that the tornado travels. In 1925, a tornado traveled 220 miles through three states. Estimate the wind speed near the center of the tornado.
Dec 92:10 PM
Evaluate Logarithms and Graph Logarithmic Functions
Dec 92:10 PM
Evaluate Logarithms and Graph Logarithmic Functions
4.4 Evaluate Logarithms and Graph Logarithmic Functions • Definition of a logarithm • Common logs • Natural logs • Inverse properties • Graphs of common and natural logs
Dec 92:10 PM
Dec 92:10 PM
5
