LOGARITHMS Section 4.2 JMerrill, 2005 Revised 2008

Exponential Functions 1111. Graph the exponential equation f(x) = 2x on the graph and record some ordered pairs.xf(x)

Review 2222. Is this a function? –Y–Y–Y–Yes, it passes the vertical line test (which means that no x’s are repeated) 3333. Domain? Range?

Review 2222. Is the function one- to-one? Does it have an inverse that is a function? –Y–Y–Y–Yes, it passes the horizontal line test.

Inverses To graph an inverse, simply switch the x’s and y’s (remember???) f(x) =f -1(x) = xf(x) xf(x)

Now graph  f(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) xf(x)

Graphing Both on the Same Graph Can you tell that the functions are inverses of each other? How?

Graphing Both on the Same Graph Can you tell that the functions are inverses of each other? How? They are symmetric about the line y = x!

Logarithms and Exponentials  The inverse function of the exponential function with base b is called the logarithmic function with base b.

Definition of the Logarithmic Function  For x > 0, and b > 0, b  1  y = log b x iff b y = x  The equation y = log b x and b y = 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: log b x = y Exponential: b y = x Exponent Base Exponent Base The 1 st to the last = the middle

Changing from Logarithmic to Exponential Form aaaa.log5 x = 2means52 = x SSSSo, x = 25 bbbb.logb64 = 3meansb3 = 64 SSSSo, b = 4 since 43 = 64 YYYYou do: cccc. log216 = xmeans SSSSo, x = 4 since 24 = 16 dddd. log255 = xmeans SSSSo, x = ½ since the square root of 25 = 5! 2 x = x = 5

Changing from Exponential to Logarithmic aaaa.122 = xmeans log12x = 2 bbbb.b3 = 9means logb9 = 3 YYYYou do: cccc. c4 = 16means dddd. 72 = xmeans log c 16 = 4 log 7 x = 2

Properties of Logarithms BBBBasic Logarithmic Properties Involving One: llllogbb = 1 because b1 = b. llllogb1 = 0 because b0 = 1 IIIInverse Properties of Logarithms: llllogbbx = x because bx = bx bbbblogbx = x because b raised to the log of some number x (with the same base) equals that number

Characteristics of Graphs TTTThe x-intercept is (1,0). There is no y-intercept. TTTThe y-axis is a vertical asymptote; x = 0. GGGGiven logb(x), If b > 1, the function is increasing. If 0<b<1, the function is decreasing. TTTThe graph is smooth and continuous. There are no sharp corners or gaps.

Transformations Vertical Shift VVVVertical shifts –M–M–M–Moves the same as all other functions! –A–A–A–Added or subtracted from the whole function at the end (or beginning)

Transformations Horizontal Shift HHHHorizontal shifts –M–M–M–Moves the same as all other functions! –M–M–M–Must be “hooked on” to the x value!

Transformations Reflections gggg(x)= - logbx RRRReflects about the x-axis gggg(x) = logb(-x) RRRReflects about the y-axis

Transformations Vertical Stretching and Shrinking ffff(x)=c logbx SSSStretches the graph if the c > 1 SSSShrinks the graph if 0 < c < 1

Transformations Horizontal Stretching and Shrinking ffff(x)=logb(cx) SSSShrinks the graph if the c > 1 SSSStretches the graph if 0 < c < 1

Domain  Because 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 Con’t. Domain

Properties of Commons Logs General Properties Common Logarithms (base 10) log b 1 = 0 log 1 = 0 log b b = 1 log 10 = 1 log b b x = x log 10 x = x b log b x = x 10 logx = x

Properties of Natural Logarithms General Properties Natural Logarithms (base e) log b 1 = 0 ln 1 = 0 log b b = 1 ln e = 1 log b b x = x ln e x = x b log b x = x e lnx = x