f(x) = a log b(x)
Like an exponential, a logarithm's essential features can be described with just two parameters.
Like an exponential, the parameter b is called the base.
Unlike an exponential, the parameter a is not the y-intercept! Indeed, members of this basic family of logarithms have no y-intercepts. To discover the meaning of a , we must consider more closely the inverse nature of exponentials and logarithms.
Recall that if
More on log notation:
Exponentials and logarithms are just different ways of expressing this relationship. The function
This explains the "mirror image" relationship between exponentials and logarithms with the same base. If (x, y) is an input-output pair for one function, then (y, x) is an input-output pair for the other.
It also helps us to explain the meaning of the parameter a . Since
It is better to think of a as a scaling factor, adjusting the outputs of logb(x) up or down as a increases or decreases, respectively.
Laws of logarithms:
Because the base of a logarithm is really the base of an exponential in disguise, we carry over the restriction given for exponentials:
The base b in a logarithmic function must be positive.
For exponentials, this condition assured that outputs from
As with exponential functions, the base is responsible for a logarithmic function's rate of growth or decay.
The key algebraic property of logarithmic functions is the following.
That is, multiplying any input x by a constant k results in adding a constant interval
The property of input multiples producing equal output steps is the basis for many logarithmic scales used in applications.
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