Exponential Functions and its Inverse
an exponential function of the form y = bx , b > 0, b not equal 1, has
a repeating pattern of finite differences
a rate of change that is increasing proportional to the function for b > 1
a rate of change that is decreasing proportional to the function for 0 < b < 1
An exponential function of the form y = bx , b > 0, b not equal 1,
has a domain X E R
a range Y E R, Y > 0
a y-intercept of 1
has a horizontal asymptote at y = 0
is increasing on its domain when b > 1
is decreasing on its domain when 0 < b < 1
The inverse of y = bx is a function that can be written as x = by.
has a domain of X E R, x > 0
a range of Y E R
a x-intercept of 1
has vertical asymptote at x = 0
is a reflection of y = bx about the line y = x
is increasing on its domain when b > 1
is decreasing on its domain when 0 < b < 1
Logarithms
a logarithmic function is the inverse of the exponential function
The value of logbx is equal to the exponent to which the base, b, is raised to produce product x
Exponnetial equations can be written in logarithmic form, and vice versa
y = b^x x = logby
y = logbX x = b^y
Exponential and logarithmic functions are defined only for positive values of the base that are
not equal to one. In other words, b not = 1, and x > 0.
The logarithm of x to base 1 is only valid when x = 1, in which base y has an infinite number
of solutions and is not a function. Common logarithms are logarithms wit ha base of 10. It is not necessary to write the base for
common logarithms: logx means log base 10 x.
Transformations of logarithmic Functions
The techniques for applying transformations to logarithmic functions are the same for those
used for other functions:
y = log x + c
translate up c units if c > 0
translate down c units if c < 0
y = log( x – d)
translate right d units if d > 0
translate left d units if d < 0
y = a log x
stretch vertically by a factor of |a| if |a| > 1
Compress vertically by a factor of |a| if |a| < 1
Reflect in the x-axis if a < 0
y = log (kx)
compress horizontally by a factor of |1/k| if |k| < 1, k not = 0.
Reflect in the y axis if k < 0.
When all transformations are combined, they follow the form:
f(x) = a log[k(x-d)] + c
Power of logarithms

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