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Properties of Logarithms. 1. log a 1 = 0 since a 0 = 1. 2. log a a = 1 since a 1 = a. 3. log a a x = x and a log a x = x inverse property. 4. If log a x = log a y , then x = y . one-to-one property. Examples : Solve for x : log 6 6 = x. - PowerPoint PPT Presentation

### Transcript of Properties of Logarithms

• Properties of Logarithm

• ExamplesExpand: 1. log3(2x) = log3(2) + log3(x)2. log4( 16/x ) = log4(16) log4(x)3. log5(x3) = 3log5(x)4. log2(8x4) log2(5) = log2(8) + log2(x4) log2(5)

• ExamplesCondense: 1. log2(x) + log2(y) = log2(xy)2. log3(4) log3(5)= log3(4/5)3. 3log2(x) 4log2(x + 3) + log2(y) log2(x3y) log2((x + 3)4)

• Natural Logarithm and eis used to denote

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• Using the natural log - lnEvaluate the following lns:Without using a calculator find the value of:

• The laws of natural logarithms

• Find x, if Remember the base of a natural log is e.Rearrange in index form.Find x in each of the following: Take a natural log of both sides.Use the power rule.

• The graph of exponential funtionThe graph of f(x) = ax, a > 1

• Example 3xy22 4Example: Sketch the graph of f(x) = 2-x. State the domain and range.Domain: (, ) Range: (0, )

• Transformation of exponential graphsExample: Sketch the graph of f(x) = 2 x + 2State the domain and range.xy24Domain: (, ) Range: (0, ) f(x) = 2 x + 2 f(x) = 2x ( 22 ) f(x) = 2x (4)

• Example 3xy22 4Example: Sketch the graph of f(x) = -2x. State the domain and range.Domain: (, ) Range: (0, -)

• Graph of f(x) = ex yx2 2 246Graph of Natural Exponential Function f(x) = exThe irrational number e, wheree 2.718281828 is used in applications involving growth and decay.

xf(x)-20.14-10.380112.7227.39

Graphs of Logarithmic Functions*The graphs of logarithmic functions are similar for different values of a. f(x) = log4 x

Graphs of Logarithmic Functions*Shifting Graph of Logarithmic Function f(x) = log4 (x-1)

Graphs of Logarithmic Functions*Shifting Graph of Logarithmic Function f(x) = 2+log4x12

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