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Math 140

4.1/4.2 – Exponential and Logarithmic Functions

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Some things just don’t grow linearly, they grow exponentially (ex: population, compound interest).

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Some things just don’t grow linearly, they grow exponentially (ex: population, compound interest).

Source: http://en.wikipedia.org/wiki/File:US_Population,_1790_-_2011.svg

U.S. Population

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ex: ex:

To model such behavior, we use the exponential function, .

is the base (, ).

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Natural Exponential Base:

(Amount of money you’d have in an account if you invested $1 at 100% interest rate per year for one year, where interest is compounded continuously.)

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In general, the continuously compounded interest formula is , and the regular compound interest formula is .

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Properties of exponents:

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Ex 1.Evaluate:

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The logarithmic function is the ______________ of the exponential function.

What does that mean?

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The logarithmic function is the ______________ of the exponential function.

What does that mean?

inverse

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The logarithmic function is the ______________ of the exponential function.

What does that mean?

inverse

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Definition

(for )

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Graphically

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Algebraically

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means means

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Ex 2.Evaluate.

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iff

Properties of logarithms

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Ex 3.Expand:

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Ex 4.Expand:

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Ex 5.Solve:

Ex 6.Solve:

Ex 7.Solve:

Ex 8.Solve:

Solving Exponential and Logarithmic Equations

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To evaluate logs with any base, you can change them to natural logs with this formula:

ex: