Exponents & Logarithms
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Exponential Algebra
Exponential Notation
Exponential expressions are a shortcut for writing the multiplication of a number by itself many times:
Here a is called the base and n the exponent. We say that a is raised to the nth power.
Note that a1=a, since we have 1×a=a.
Exponent with zero
Any number raised to the power zero is
This also applies, somewhat confusingly, when a=0:
Multiplying exponents
When multiplying exponentials with the same base, the following rule applies:
Exponential of exponential
An exponential can be the base of another exponential:
Exponents of products & quotients
When exponentials with the same power are being multiplied or divided, the bases can be combined:
Logarithm algebra
Definition of the logarithm
Logarithms are a mathematical tool for asking "what power of a given base gives a specific value". We write this as
Here, a is called the base, and must be positive. b must also be positive. The value of x, however, can be any real number.
log base 10
In science and mathematics it is so common to use log10 that we can simply write the shorthand log to indicate log10.
For example, log(0.001)=−3 since 10−3=0.001.
Natural logarithm
Another special logarithm is the one in base e. We call it the natural logarithm due to fundamental importance of e across mathematics.
For example, ln(e3)=3.
Using logs to solve exponential equations
Logarithms can be used to solve exponential equations:
Radicals and Roots
nth Roots
For any number a and whole number n,
is called the nth root of a.
The nth root of a is the number that gives you a when raised to the nth power:
Note that this root is positive when n is even.
Exp & Log functions
Exponential functions
An exponential function has the form f(x)=ax for some base a>0. The domain of f is R, and the range is f(x)>0:
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Graphing Exponential Functions
In general, to graph an exponential function of the form f(x)=cax+k, find the y-intercept of the curve, then analyze the behavior of the function on both ends (as x→∞ and as x→−∞). If possible, plotting other easily calculated points - often f(1) or f(−1).
The y-intercept is at (0,c+k) because f(0)=ca0+k=c(1)+k.
On one end, the curve will approach y=k.
For a<1, as x→∞, f(x)→c(0)+k.
For a>1, as x→−∞, f(x)→c(0)+k.
On the other end, the curve will rise with increasing steepness.
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Exponential growth
Exponential growth describes quantities that increase by the same factor over a certain amount of time. Algebraically, exponential growth are functions of the form
where b>1. b is called the growth factor.
Note: Aekt is another model for exponential growth if the instantaneous growth rate, k, is positive.
Stewart EJ, Madden R, Paul G, Taddei F (2005), CC BY-SA 4.0
Exponential decay
Exponential decay describes quantities that decrease by the same factor over a certain amount of time. Exponential decay are functions of the form
where 0<b<1. b is called the decay factor.
Note: Aekt is another model for exponential growth if the instantaneous growth rate, k, is negative.
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Logarithmic functions
A logarithmic function has the form f(x)=logax, for a>0. The domain of f is x>0, and the range is R:
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