Chapter 3: Exponential and Logarithmic FunctionsStudy Guide |
Section 3.1
Hints:
1. Functions whose equations contain a variable in the
exponent are called exponential functions.
Ex.:
2. If the base of the exponential function has a base that is greater than one, the graph looks like the example below.
3. If the base of the exponential function has a base that is between zero and one, the graph looks like the example below.
4. The characteristics of exponential functions of the form are given on page 359.
5. The irrational number, symbolized by the letter e, appears as the base in many application functions. The value of e is approximately 2.71828. . . The graph of the function, , is between the graphs of and , since e is between 2 and 3.
6. Compound interest formulas are one common application of exponential functions.
is the formula for n compounding per year.
is the formula for continuous compounding.
Section 3.2
Hints:
1. The inverse of the exponential function with a base b
is a logarithmic function with base b.
The logarithmic function is defined as
2. It is very important for you to be able to convert an
exponential form, ,
to a logarithmic form, ,
and vice versa.
3. Since logarithms are exponents, they have properties that can be verified using properties of exponents. Some of the basic properties of logarithms are as follows.
1)
2)
3)
4)
4. The figure below shows a graph of an exponential function and its inverse (a logarithmic function) when b>1.
The figure below shows a graph of an exponential function and its inverse (a logarithmic function) when 0<b<1.
Characteristics of the graphs of logarithmic functions are given on page 389.
Common logarithms can be found using a calculator. Properties of common logarithms are given on page 374.
6. Natural logarithms are logarithms with base e. The notation used for a natural logarithm is "ln".
Ex.:
Natural logarithms can be found using a calculator. Properties of natural logarithms are given on page 376.
Section 3.3
Hints:
When writing the sum of logarithms the order of the terms does not matter; but, when writing the difference of logarithms, watch the order of the terms. You must subtract the logarithm of the denominator of the quotient from the logarithm of the numerator of the quotient.
4. When you use the product, quotient, or power rule on a logarithmic expression, you are expanding the expression. Think of the word "expand" as "to make larger". When you use these properties, your final expression will be larger than your initial expression. Sometimes it is necessary to use more that one property to completely expand the given logarithmic expression.
5. When you write the sum or difference of two or more logarithmic expressions as a single logarithmic expression, you are condensing a logarithmic expression. Think of the word "condense" as "to make smaller". When you use these properties, your final expression will be smaller than your initial expression. The properties for condensing logarithms are the same as the properties for expanding. The only difference is that sides of each property are reversed. Compare the expanding properties on page 383 with the condensing properties on page 384.
6. A calculator will only find common logarithms and natural logarithms. You can use the change-of-base property to find a logarithm with any other base.
Section 3.4
Hints:
1. An exponential equation is an equation containing a variable
in an exponent. You may use either natural or common logarithms and your
calculator to solve exponential equations. The steps for solving this type
of equation using natural logarithms are
1) Isolate the exponential expression.
2) Take the natural logarithm on both sides of the equation.
3) Simplify using one of the following properties:
or
4) Solve for the variable.
3. Logarithmic expressions are defined only for
logarithms of positive real numbers. You must always check
your solutions in the original equation. You should exclude
any solution that produces the logarithm of a negative number or the logarithm
of 0.
Section 3.5
Hints: