Exponential

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Exponential Logarithmic Functions
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• The exponential function f with base a is
  denoted by f(x)ax, where a?1 , and x is any
  real number.
• The function value will be positive because a
  positive base raised to any power is positive.

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• Ex if the base is 2 and x 4, the function
  value f(4) will equal 16. The graph of f(x)2x
  would be (4, 16).

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Exponential functions Definition Take a gt 0 and
not equal to 1 . Then, the function defined by
f R -gt R x -gt ax is called
an exponential function with base a.
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Graph and properties Let f(x) an exp. fun. with
a gt 1.Let g(x) an exp. Fun. with 0 lt a lt 1.
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• From the graphs we see that
• The domain is R
• The range is the set of strictly
• positive real numbers
• The function is continuous in its domain
• The function is increasing if a gt 1 and
  decreasing if 0 lt a lt 1
• The x-axis is a horizontal asymptote

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Logarithmic functions Definition and basic
properties Take a gt 0 and not equal to 1 . Since
the exponential function f R -gt R x -gt ax are
either increasing or decreasing, the inverse
function is defined. This inverse function is
called the logarithmic function with base a. We
write loga (x)
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for x gt 0 we have aloga(x) x for all x we have
loga(ax) x
Graph Let f(x) a logarithmic function with
a gt 1.Let g(x) a logarithmic
function with 0 lt a lt 1.
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log(x.y) log(x) log(y)
log(x/y) log(x) - log(y) log(xr ) r.log(x)





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Pf log(x.y) u then au x.y (1) Let
log(x) v then av x (2) Let log(y) w
then aw y (3) From (1) , (2) and (3) au
av . aw gt au av w gt u v w
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Change the base of a logarithmic fun.
Theoremfor each strictly positive real number a
and b, different from 1,
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• Definition of Logarithmic Function
• For x gt0, agt0 , and a ? 1, we have
• f(x)loga(x) iff a f(x) x
• Since x gt 0, the graph of the above function
  will be in quadrants I and IV.

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• Comments on Logarithmic Functions
• The exponential equation 4364, could be written
  in terms of a logarithmic equation as
  log4(64)3.
• The exponential equation 5-21/25 can be
  written as the logarithmic equation log5(1/25)-2.

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• Logarithmic functions are the inverse of
  exponential functions. For example if (4, 16) is
  a point on the graph of an exponential function,
  then (16, 4) would be the corresponding point on
  the graph of the inverse logarithmic function.

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The derivatives of the logarithmic functions
d/dx logb(x) 1 / x ln (b)
d/dx ln (x) 1/x since ln e 1
Derivative of bx and ex (d/dx) bx bx ln(b)
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Ex d/dx 2x(4 x ) 2(4 x ) 2x(4 x ) ln4
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Ex d/dx ln (x2 2x -1) Ex d/dx ln (3x
2) Ex d/dx log 3 (x) 1 / x ln (3)