# LOGARITHMS Section 4.2

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LOGARITHMSSection 4.2JMerrill, 2005Revised 2008

Exponential Functions1. Graph the exponential equation f(x) = 2x on the graph and record some ordered pairs.

xf(x)01122438

Review2. Is this a function?Yes, it passes the vertical line test (which means that no xs are repeated)

3. Domain?

Range?

Review2. Is the function one-to-one? Does it have an inverse that is a function?Yes, it passes the horizontal line test.

InversesTo graph an inverse, simply switch the xs and ys (remember???)

f(x) = f -1(x) =

xf(x)01122438

xf(x)10214283

Now graphf(x)

f-1(x)

How are the Domain and Range of f(x) and f -1(x) related? The domain of the original function is the same as the range of the new function and vice versa.

f(x) = f -1(x) =

xf(x)01122438

xf(x)10214283

Graphing Both on the Same GraphCan you tell that thefunctions are inversesof each other? How?

Graphing Both on the Same GraphCan you tell that thefunctions are inversesof each other? How?

They are symmetricabout the line y = x!

Logarithms and ExponentialsThe inverse function of the exponential function with base b is called the logarithmic function with base b.

Definition of the Logarithmic FunctionFor x > 0, and b > 0, b 1y = logbx iff by = x

The equation y = logbx and by = x are different ways of expressing the same thing. The first equation is the logarithmic form; the second is the exponential form.

Location of Base and Exponent

Logarithmic: logbx = y

Exponential: by = xExponentBaseExponentBaseThe 1st to the last = the middle

Changing from Logarithmic to Exponential Forma.log5 x = 2 means52 = xSo, x = 25b.logb64 = 3meansb3 = 64So, b = 4 since 43 = 64You do:c. log216 = xmeansSo, x = 4 since 24 = 16d. log255 = x meansSo, x = since the square root of 25 = 5!2x = 1625x = 5

Changing from Exponential to Logarithmica.122 = xmeans log12x = 2b.b3 = 9means logb9 = 3

You do:c. c4 = 16meansd. 72 = xmeanslogc16 = 4log7x = 2

Properties of LogarithmsBasic Logarithmic Properties Involving One:logbb = 1 because b1 = b.logb1 = 0 because b0 = 1

Inverse Properties of Logarithms:logbbx = x because bx = bxblogbx = x because b raised to the log of some number x (with the same base) equals that number

- Characteristics of GraphsThe x-intercept is (1,0). There is no y-intercept.The y-axis is a vertical asymptote; x = 0.Given logb(x), If b > 1, the function is increasing. If 0
TransformationsVertical Shift Vertical shiftsMoves the same as all other functions!Added or subtracted from the whole function at the end (or beginning)

TransformationsHorizontal ShiftHorizontal shiftsMoves the same as all other functions!Must be hooked on to the x value!

TransformationsReflectionsg(x)= - logbxReflects about the x-axisg(x) = logb(-x)Reflects about the y-axis

TransformationsVertical Stretching and Shrinkingf(x)=c logbxStretches the graph if the c > 1Shrinks the graph if 0 < c < 1

TransformationsHorizontal Stretching and Shrinkingf(x)=logb(cx)Shrinks the graph if the c > 1Stretches the graph if 0 < c < 1

DomainBecause a logarithmic function reverses the domain and range of the exponential function, the domain of a logarithmic function is the set of all positive real numbers unless a horizontal shift is involved.

Domain Cont.Domain

Domain

Domain

Properties of Commons Logs

General PropertiesCommon Logarithms(base 10)logb1 = 0 log 1 = 0 logbb = 1 log 10 = 1 logbbx = x log 10x = x blogbx = x 10logx = x

Properties of Natural Logarithms

General PropertiesNatural Logarithms(base e)logb1 = 0 ln 1 = 0 logbb = 1 ln e = 1 logbbx = x ln ex = x blogbx = x elnx = x