Chapter 8
Exponents, Logarithms, and Inverse Functions
By Boundless
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To find the inverse function, switch the
Functional composition allows for the application of one function to another; this step can be undone by using functional decomposition.
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Domain restriction is important for inverse functions of exponents and logarithms because sometimes we need to find an unique inverse.
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A composite function represents, in one function, the results of an entire chain of dependent functions.
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Logarithmic functions and exponential functions are inverses of each other. That is, they undo each other.
The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number.
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Any positive number can be used as the base of a logarithm but certain bases (
Logarithmic and exponential forms are closely related, and an equation in either form can be freely converted into the other.
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The exponential function
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Logarithmic functions can be graphed manually or electronically with points generally determined via a calculator or table.
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Some functions with rapidly changing shape are best plotted on a scale that increases exponentially, such as a logarithmic graph.
A useful property of logarithms states that the logarithm of a product of two quantities is the sum of the logarithms of the two factors. In symbols,
A simplifying principle of logarithms is that the logarithm of the
The logarithm of the ratio of two quantities is the difference of the logarithms of the quantities. In symbols,
A logarithm written in one base can be converted to an equal quantity written in a different base.
The number
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The function
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The natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828.
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Population can fluctuate positively or negatively and can be modeled using an exponential function.
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A realistic model of exponential growth must dampen when approaching a certain value. This limited growth is modelled with the logistic growth model:
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Compound interest is accrued when interest is earned not only on principal, but on previously accrued interest: it is interest on interest.
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Exponential decay is the result of a function that decreases in proportion to its current value.