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LOGARITHMS
- Simplifying Logarithms…………………………………………
- Properties of Logarithms…….…………………………….……
- Expanding Logarithmic Expressions…….……………….…….
- Condensing Logarithmic Expressions…….....………..………..
- Practice Using Properties of Logarithms…………………..…..
- Base Change Formula……………………………………..…….
- Solving Logarithmic Equations…….……………………...…… - Solving Exponential Equations…………………………………
- Finding the Domain of a Logarithmic Function….……..……..
- Finding the Vertical Asymptote of a Logarithmic Function….
- Graphing Logarithmic Functions.……………………….……..
- Finding the Inverse of a Function…….…………………..…….
- Interest Formulas……………………………………….……….
- Word Problems………….……………………………...………..
Objectives
The following is a list of objectives for this section of the workbook.
By the time the student is finished with this section of the workbook, he/she should be able to…
- Evaluate a simple logarithm without the aid of a calculator.
- Express a logarithmic statement is exponential form.
- Express a statement in exponential form in logarithmic form.
- Expand a logarithmic expression as the sum or difference of logarithms using the properties of logs.
- Condense the sum or difference of logarithms into a single logarithmic expression.
- Evaluate logarithms using the base change formula.
- Solve logarithmic equations.
- Evaluate the solution to logarithmic equations to find extraneous roots.
- Solve equations with variables in the exponents.
- Find the range and domain of logarithmic functions.
- Graph a logarithmic function using a table.
- Find the inverse of a function.
- Verify two functions are inverses of each other.
- Identify a one-to-one function.
- Use the compound interest formulas.
Math Standards Addressed
The following state standards are addressed in this section of the workbook.
Algebra II
11.0 Students prove simple laws of logarithms.
11.1 Students understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents. 11.2 Students judge the validity of an argument according to whether the properties of real numbers, exponents, and logarithms have been applied correctly at each step.
13.0 Students use the definition of logarithms to translate between logarithms in any base.
14.0 Students understand and use the properties of logarithms to simplify logarithmic numeric expressions and to identify their approximate values.
15.0 Students determine whether a specific algebraic statement involving rational expressions, radical expressions, or logarithmic or exponential functions is sometimes true, always true, or never true.
24.0 Students solve problems involving functional concepts, such as composition, defining the inverse function and performing arithmetic operations on functions.
Simplifying Logarithms
Evaluate each of the following logarithms without the use of a calculator.
A) log 81 3 = B) (^) 4
log 2
= C) log 12 144 = D) (^) 6
log 36
E) 2
3
log 4
= F) log (^) 0.25 4 = G) log 3 − 3 = H) log 4 8 =
I) 81
log 27
= J) 1
16
log 32 = K) log 0 4 = L) log 10 1 =
M) 4
log 8
= N) 27
log 3
= O) log 3 9 = P) log 6 6 3 x^ =
Q) 36
log 6
= R) log 128 2 = S) (^) 1 4
log 16 = T) log (^) z z^2 x =
U) ln e^12 = V) 3 log 5 3 = W) ln1 = X) e ln 4^ x =
Y) log 16 2 2 = Z) log 3 59 = a) log 9 3 33 = b) (^) (^5 )
log 25
c) (^) 53 6
log 25
= d) ln 5 x^2 e =
Properties of Logarithms
The following properties serve to expand or condense a logarithm or logarithmic expression so it can be worked with.
Properties of logarithms Example
log log log
log log log
log log
a a a
a a a n a a
mn m n m m n n m n m
4 4 4
2 2 2 3 3 3
log 3 log 3 log 1 log log 1 log 5 5 log 2 1 3log 2 1
x x x x
x x
Properties of Natural Logarithms Example
ln ln ln
ln ln ln
ln n ln
mn m n m m n n m n m
3
ln 1 5 ln 1 ln 5
ln ln ln 2 2 ln 7 3ln 7
x x x x x x
These properties are used backwards and forwards in order to expand or condense a logarithmic expression. Therefore, these skills are needed in order to solve any equation involving logarithms. Logarithms will also be dealt with in Calculus. If a student has a firm grasp on these three simple properties, it will help greatly in Calculus.
Expanding Logarithmic Expressions
Write each of the following as the sum or difference of logarithms. In other words, expand each logarithmic expression.
A)
3 2 2 5
log x y z
B) log 5 3 3 xy^2
C) ( ) ( )
4 3 2 log x + 1 x − 2 D)
2 (^5 )
log 11
x y z
Condensing Logarithmic Expressions
Rewrite each of the following logarithmic expressions using a single logarithm. Condense each of the following to a single expression. Do not multiply out complex polynomials. Just
leave something like ( )
3 x + 5 alone.
A) 3log 4 x − 5log 4 y + 2 log 4 z B)
2 log log 2
x + y
C)
log 6 log log 3 3 3
log 16 log 2 log 4 3
− x − y
E) 3log 5 x + 2 log 5 y + log 5 z + 2 F) (^) 2 2
log log 3 3 3
x + y −
G) log 3 ( x + 2 ) + log 3 ( x − 2 ) − log 3 ( x + 4 ) H) ( ) ( ) ( )
log 1 log 2 log 5 3 3 3
x + + x − − x +
I) ( ) ( )
2 ln 3 ln ln 2 1 3
x + + x − x − J) ln ( x + 3 ) − ln 2( x + 5 ) + 2 ln ( x − 1 )
K) ( ) ( )
ln 3 2 ln 1 3ln 2
x + + x − − x L)
2 ln 3 6 ln ln 27 3
M)^3
1 log 2
N) ln x − 2 ln ( x + 2 ) + ln ( x − 2 )
Practice Using Properties of Logarithms
Use the following information, to approximate the logarithm to 4 significant digits by using the properties of logarithms.
log (^) a 2 ≈ 0.3562, log 3 a ≈ 0.5646, and log 5 a ≈0.
A)
log a (^) 5 B) log 18 a C) log 100 a
D) log 30 a E) log (^) a 3 F) log (^) a 75
G)
log a (^) 9 H) log (^) a^315 I) log 54 a^2
Base Change Formula
Up to this point calculators have not been used to evaluate logarithms. Remember, the logs on all calculators are base 10. If a calculator is used to evaluate a log with any other base, it will not give the correct estimate unless a specific formula is used. The following base change formula will be used to evaluate any log with a base other than 10.
log log a log
b b a
The terms log a and log b will be evaluated with a base 10, which means a calculator can now
be used. Of course natural logs are always base e , so there is no need for a formula with natural logs. Each calculator is different. Be aware of what keystrokes are needed for your particular calculator, whether the log must be entered first, or the number first. Be very careful about grouping symbols when entering these types of problems into the calculator.
Using a calculator, evaluate each of the following. Round all answers to three decimal places.
A) log 12 3 B) log 17 6 C) (^) 3
log 5
D) log 8 4
E) 6
log 12
F) log 7 ( − 35 ) G) ln14 H) ln 3.
I)
ln 2
J) ln 0 K) ln e L) ln 6.
Why can’t you take the log of a negative number?
Solving Logarithmic Equations
Here we will solve some logarithmic equations. There are a couple of ways to solve a logarithmic equation. One method is to get all logs to the left side of the equal sign and condense the problem so there is only one log. The next step would be to write the problem in exponential form and solve.
Another method involves one of the properties of simple logarithms. In particular, the property that states: “ If log (^) a x = log ay then x = y .” If the equation can be manipulated to
resemble this property, take what is inside of each and set the two terms equal to each other. This follows the same principal as exponential equations where the bases are identical. If an
exponential equation states 2 3 x +^4 = 27 ; it stands to reason, because of the equal sign, that 3 x + 4 = 7. The problem would then be solved from this point. With logarithmic equations,
the problem log 4 ( 8 x − 5 )= log 12 4 can be written as 8 x − 5 = 12.
Always check for extraneous roots!!!
Solve each of the following logarithmic equations. (Round any solutions with decimals to three decimal places)
A) log 4 x + 6 = 8 B) 2 log 3 x − 4 = 1 C) 5 + 2 log 4 x = 4
D) log 3 x − log 2 3 = 5 E) log 2( x − 1 ) = 0 F) log 4( x + 8 ) = 2
G) log 3 ( x + 5 ) + log 3 ( x + 3 )= log 35 3 H) log ( x + 3 ) + log ( x − 4 )=log 30
Solving Exponential Equations
You will now be solving exponential equations. Logarithms allow you to solve equations with variables in the exponent. We have dealt with these types of problems before, however, in the previous examples the bases of the problem could be made to match making solving the problem as simple as setting the two exponents equal to each other and solving for the variable. When ever you come to a problem with the variable in the exponent, and you cannot get the bases to match, you will need to solve the problem using logarithms. You will be using all three properties of logarithms to simplify and eventually solve the problem. You can take either the log or natural log of both sides, it makes no difference unless e is in the problem. In that case, you must use natural logs to solve the equation.
Solve each of the following exponential equations. Round solutions to three decimal places.
A) 32 x^ = 5 B) 64 x^ = 300 C) 123 x +^1 = 72
D) 5 x +^2 = 35 E) 45 x^ +^8 = 8 x −^1 F) 93 x^ +^2 = 27 x +^8
G) 123 x^ −^2 = 85 x +^1 H) 56 x^ −^5 = 62 x +^1
I) 4 e^3 x = 40 J) 1200 ex = 900 K) 3 e 4 x = 12
L) e^2^ x^ − 3 ex + 2 = 0 M) e^2 x^ − 5 ex + 6 = 0 N) e^2^ x^ + 2 ex − 24 = 0
O) 2 e^2 x^ + ex − 10 = 0 P) 52 x^ − 7 5⋅ x + 10 = 0 Q) 32 x^ + 5 3⋅ x − 14 = 0
R) (^) 8 ( 4 ( 6 ))
log log log 3
x = S) log 2 ( log 4 x )= 1 T) log 3 ( log 27 x )= − 1
Finding the Vertical Asymptote of a Logarithmic Function
f ( x ) = a log n ( bx + c ) + d f ( x ) = a ln( bx + c )+ d
To find the vertical asymptote of a logarithmic function, set bx+c equal to zero and solve. This will yield the equation of a vertical line. In this case, that vertical line is the vertical asymptote.
Example
Find the vertical asymptote of the function f ( x ) = log 3 ( 4 x − 3 ) − 2.
x x
x
Setting the bx+c term equal to zero results in this equation. Simply solve for x, and we now have the vertical asymptote of the function.
Find the vertical asymptote of each of the following logarithmic functions.
A) f ( x ) = log 5 x + 2 B) f ( x ) = log 3 ( 4 x − 1 ) − 2 C) f ( x ) = −log 3 2 x
D) f ( x ) = log 2 ( 5 − x ) E) f ( x ) = log 5 ( − x )+ 5 F) f ( x ) = ln x − 4
G) f ( x ) = ln 2( − 3 x ) H) f ( x ) = log 3 ( x + 5 ) + 1 I) f ( x ) = ln ( x − 3 )
J) f (^) ( x ) = log 2 x K) f ( (^) x ) = log 3 x L) f (^) ( x ) = − ln 5 x + 1
Graphing Logarithmic Functions
f ( x ) = a log n ( bx + c ) + d f ( x ) = a ln( bx + c )+ d
The parent functions for both logarithmic and natural logarithmic functions look almost identical. The following functions will be graphed by setting up a table. When faced with the following function: f ( (^) x ) = log 3 x , find the domain of the function first by evaluating
bx + c > 0. This is done to ensure that only appropriate values are substituted for x. The values that should be used for x are powers of the base of the logarithm such as 1 1 , , 1, 3, 9. 9 3
and The values used to substitute for x are completely dependant on the
base of the particular logarithm in the problem. When graphing a natural log with a table, any number in the domain of the function may be used. Using a calculator, get the decimal estimates for the y values of the function. DO NOT FORGET to plug in fractional powers for both logs and natural logs. The fractional powers will give the tail end of the function. Be sure to plug in the fractions that are right next to the vertical asymptote of the function. Note in the following picture the graph does not cross the y axis. That is because, in this particular example, x = 0 is the vertical asymptote. By now, you should be aware that the equation x = 0 is the equation for the y axis. Remember, always find the vertical asymptote of any logarithmic function when graphing. This must be done to ensure that it is not crossed. The range of any normal logarithmic function is all real numbers. The only time this will not be is the case is that of example K in the finding the domain section on the previous page.
Remember, most people forget the tail end of the function. Do not forget it. Make sure you note the rise of the function. The function makes a very slow gradual rise. It is not steep.
Normally, we will be graphing these functions by means of translations based on the parent function. This graphing portion is meant to give you practice at using your knowledge about logarithms and how to use powers to graph the functions. There will be an entire section devoted to nothing but functions later on in the workbook.
C) f (^) ( x ) = − log 2 x x f ( x )
D) f ( x ) = log 3 ( x + 1 ) x f ( x )
E) f (^) ( x ) = ln x + (^3) x f ( x )
F) f ( x ) = ln ( x + 2 ) x f ( x )
