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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
And since any number multiplied by 0 is 0:
When n=0, we have 00, which is technically undefined, but in most contexts is defined to be
Multiplying powers with the same base
When multiplying exponentials with the same base, the following rule applies:
Exponential of exponential
An exponential can be the base of another exponential:
Negative exponents
Dividing exponents with the same base
In general,
Exponents of products & quotients
When exponentials with the same power are being multiplied or divided, the bases can be combined:
Exponential Equations (Equating Indices)
If two exponentials in the same positive base are equal, their exponents must be equal:
Exponentials can also appear in equations with one or more unknown:
Now we can equate the exponents:
Radicals and Roots
nth Roots
For any number a and positive integer 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:
If n is even, then (n√a)n is necessarily positive, so we must restrict a>0.
Roots of negative numbers
If a is negative, n√a is negative for all odd n.
For even n, no real n√a exists.
Converting nth roots to fractional exponents
Roots can always be written as fractional exponents and vice versa:
Rational exponents
Utilizing nth roots and exponential laws we can rewrite any rational exponent:
Simplest form radicals
A radical is in simplest form if the integer under the radical sign is as small as possible.
For example, the simplest form of √48 is 4√3. We can simplify by splitting the radical into a reducible and irreducible part:
Simplest form fractions with radicals (multiplying by roots)
A fraction in simplest form does not have a radical in the denominator.
For a fraction of the form √ba where a∈Z,b∈N, we find the simplest form by mutliplying the numerator and denominator by √b:
When we remove a radical from a denominator, we call it rationalizing the denominator.
Rationalizing Denominators with Conjugates
To simplify a fraction of the form b+√ca, multiply the fraction by b−√cb−√c.
b−√c is called the conjugate of b+√c.
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 it must be positive and not equal to 1. b must also be positive. The value of x, however, can be any real number.
Evaluating logs algebraically
Some logarithms can be evaluated by hand using the fact that
For example, we can find log279 by solving the equation
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 the fundamental importance of e across mathematics.
For example, ln(e3)=3.
Evaluating logs using technology
If a and b are not powers of the same base, the log cannot be easily computed by hand. But we can use a calculator to evaluate them approximately.
Sum and difference of logs
The sum of logs with the same base is the log of the products:
We have a similar rule for the difference of logs:
Log power rule
Log change of base
We can change the base of a logarithm using the law
for any choice of positive b=1.
Using logs to solve exponential equations
Logarithms can be used to solve exponential equations:
Exp & Log functions
Exponential functions
An exponential function has the form f(x)=ax for some base a>0 (and a=1). 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 is modeled by 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 is modeled by functions of the form
where 0<b<1. b is called the decay factor.
Note: Aekt is another model for exponential decay 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>1. The domain of f is x>0, and the range is R:
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Log and exponent functions are inverses
The functions logax and ax are inverses:
This can be seen by the symmetry of their graphs in the line y=x:
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