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Every exponential function f(x) = ax, with a > 0 and a ≠ 1, is a one-to-one function by the Horizontal Line Test and, therefore, has an inverse function.
The inverse function f –1 is called the logarithmic function with base a and is denoted by loga.
Recall from Section 2.7 that f –1 is defined by: f –1(x) = y f(y) = x
This leads to the following definition of the logarithmic function.
Graphs of Logarithmic Functions
The graph of f –1(x) = logax is obtained by reflecting the graph of f(x) = ax in the line y = x.
The figure shows the case a > 1.
Let a be a positive number with a ≠ 1.
The logarithmic function with base a, denoted by loga, is defined by:
logax = y ay = x
So, logax is the exponent to which the base a must be raised to give x.
The logarithm with base 10 is called the common logarithm.
It is denoted by omitting the base: log x = log10x
By the definition of inverse functions, we have: log x = y 10y = x
The logarithm with base e is called the natural logarithm.
It is denoted by ln: ln x = logex
By the definition of inverse functions, we have: ln x = y ey = x
When we use the definition of logarithms to switch back and forth between the logarithmic form logax = y and the exponential form
ay = x it’s helpful to notice that, in both forms, the base is the same:
a is the base.
logax = y
ay = x
y is the exponent.
Properties of Logarithms
loga1 = 0
We must raise a to the power 0 to get 1.
logaa = 1
We must raise a to the power 1 to get a.
logaax = x
We must raise a to the power x to get ax.
alogax = x
logax is the power to which a must be raised to get x.
Properties of Natural Logarithms
ln 1 = 0
We must raise e to the power 0 to get 1.
ln e = 1
We must raise e to the power 1 to get e.
ln ex = x
We must raise e to the power x to get ex.
eln x = x
ln x is the power to which e must be raised to get x.
log101000 = .
log232 = .
log164 = .
Find the domain of the function, and sketch the graph.
(a) g(x) = 2 + log5(x - 3)
Find the domain of the function
f(x) = ln(x +2)
Scientists model human response to stimuli (such as sound, light, or pressure) using logarithmic functions.
For example, the intensity of a sound must be increased manyfold before we “feel” that the loudness has simply doubled.
The psychologist Gustav Fechner formulated the law as
S is the subjective intensity of the stimulus.
I is the physical intensity of the stimulus.
I0 stands for the threshold physical intensity.
k is a constant that is different for each sensory stimulus.