LOGARITHMS

Objectives:

• To learn what a logarithm does and how to use it.
• Evaluate a logarithmic expression.

Related topics: 1-4, 7, 9, 10, 13, 25.

Recall 1:  You may have forgotten the basic rules for manipulating exponents.  Here are some of those rules.

1.1 There is a particular number, denoted e, which is used extensively in science and mathematics.  Its value is approximately 2.718 and may be found as the "e" button on most calculators.  Computers generally expect you to enter the number e as Exp(1).  Therefore, if you wish to enter the number e^4 on a computer, you would type Exp(4).  In the computer algebra system Mathematica, you would type Exp[4].  Note the capital letter and the square brackets around the 4.

1.2 To enter mathematical operations of multiplication and exponentiation on a computer, use the * for multiplication and ^ for exponentiation.  Therefore,

3*4 = 12 and 3^4 = 81.

1.3 You may wish to think of the exponentials a^x and e^x as functions of x.  You may enter e^x in Mathematica as Exp[x].

The rules for Exp[x] are then expressed as

Exp[x+y] = Exp[x] *Exp[y]

Exp[x]^y = Exp[xy]

Exp[x-y] = Exp[x]/Exp[y]

Exp[0] = 1

Recall 2: If a is different from 0, then a^x is different from 0 for all x.

Recall 3: When two functions are connected by the properties y = f(x) and x = g(y), we say that f(x) is the inverse function of g(y) and g(y) is the inverse function of f(x).  Of course this also means that g(f(x)) = x and f(g(y)) = y.

Recall 4: If you plot the graphs of a function and its inverse you will find that they are mirror images of each other when the imaginary mirror is placed along the diagonal line y = x.  See illustration below.

The illustration below shows the functions Exp[x] and its inverse.

Introduction 1 (Logarithms):

The inverse function of a^x is called the logarithm to the base a and is denoted

The inverse function of e^x is called the natural logarithm of x and is often denoted Ln(x).

The graphs of e^x and Ln(x) are displayed in the illustration above.  To enter a logarithm to the base a in Mathematica type Log[a,x].  If you type Log[x], Mathematica will interpret the base as e.  Most calculators have buttons labeled "log" and "ln" for Log10(x) and Ln(x), respectively.

Recall 5 As you would expect, the rules for manipulating logarithms look very much like the reverse of those for manipulating exponents.  They are:

 Rules of Logarithms Let a be a positive number such that a does not equal 1, let n be a real number, and let u and v be positive real numbers.

Important Note:  The domain of the logarithm function is the set of all positive numbers.  Therefore, Log(x) makes no sense when x is a negative number.

Recall 6  Since the logarithm and exponential functions are inverses of each other, we have the following relationships:

Recall 7  One of the many great uses of the logarithm function is to reduce exponents to products.  So, for example, if you are ever faced with an equation like 4 = 2^(x-3) and you need to know what x is, all you have to do is take the logarithm to the base 2 of both sides of the equation to get

Log2(4) = Log2[2^(x-3)]

and you can proceed by knowing that the left side is equal to 2 and the right side is equal to x - 3.  Then all you have to do is solve the equation 2 = x - 3 to get x = 5.

Recall 8 You know that a^b = a^c if and only if b = c.  Therefore you may solve equations such as

because you would know that x^2 + 6 = -x, and that would mean

x^2 + x + 6 = 0,  or

(x + 3)(x + 2) = 0.

This means that x = -3 or x = -2.

Recall 9 You know that Logab = Logac if and only if b = c.  Therefore you may solve equations of the form

Log2(x^2 - 6) = Log2(x) by what we just said in the last paragraph.

From Recall 8, you know that

From this it follows that x^2 - 6 = x and therefore x = 3 or x = -2.  But x cannot equal -2 because when we check it we find that we have to make sense of Log (-2).  The logarithm function does not have negative numbers in its domain.  (See the important note above.)  So the only solution is x = 3.

Recall 10 If you know the logarithm to the base "a" of a number then you know it to any other base by using the conversion formula below:

Calculators often only have two kinds of log buttons, log and ln.  If you wish to find Log2(x) then all you have to do is find the quotient ln(x)/ln(2).  If you are simply converting from common logarithms (base 10) to natural logarithms (base e) or back, you could use the conversion factors given by

Ln(x) = 2.30259 Lg(x), or Lg(x) = .43429 ln(x).

Examples

Example 1.

Example 2.

Example 3.      The expression

can be written as a single logarithm

Example 4.      To calculate Log5(1/3)  you may have to calculate the ratio -Ln(3)/Ln(5)

Example 5.      To solve an equation Log2(x - 4) = 1 - Log2(x - 3) you would collect the logarithms to one side to get Log2(x - 4) + Log2(x - 3) = 1, then group the logarithms together to get Log2((x - 4)(x - 3)) = 1.  Now recall that if Loga(y) = 1, then y = a.  Therefore (x - 4)(x-3)=2, which gives x = 2 or x = 5.  But x = 2 cannot be a solution because Log2(x - 4) is the logarithm of a negative number when x = 2 and there is no such thing as the logarithm of a negative number.  So the only solution is x = 5.

Example 6.      What if you want to find x when Log2(x - 4) + Log2(x - 3) = 2, then collect the logarithms to get Log2((x - 4)(x - 3)) = 2.  Now use the fact that a^b = a^c if and only if b = c to see that

Therefore,  (x - 4)(x - 3) = 4.  If you multiply out the left side you get x^2 - 7x - 12 = 4.  The equation then becomes x^2 - 7x - 16 = 0.  This equation does not factor; so we have to use the quadratic formula to get the two solutions.

But one of the solutions is less than 3 and therefore is not in the domain of Log2(x - 3).  So we have only one valid solution.

EXERCISES

Note: log(x) is a shorthand notation for the logarithm to the base 10 of x.

1. Find

2.  If Log4(x) = 5/2, what is x?

3  Simplify Log5(5)

4 Simplify Exp[ln7].

5 Write Log3(x + 2) + 2Log3x - Log32 as a single logarithm

6 Use natural logarithms to evaluate Log527

7 If Loga(8/27) = 3, what is a?

8 Solve the equation 9^x = 12 for x.

9 Solve the equation 7^(2x-1) = 7^(x+2) for x.

10. Solve the equation 4^(2x-1) = 3^(x+2) for x.

11 Solve the equation   exp(x^2)=100     for x.

12 Solve the equation 5 = 2(3 - e^x) for x.

13 Solve the equation Logx(x + 6) - Logx(x + 2) = Logxx for x.

14 Solve the equation Log3(3x^2)^2  - 1 = 3 for x.

15 If y = Log2(x^2 + 6), write x as a function of y.

16 Solve the equation (Logx)^2 = Log(x^2) for x.

1.  6

2.  x = 32

3.  1

4.  7

5.  Log3[x^2(x+2)/2]

6. Ln(27)/Ln(5)

7.  a = 2/3

8.  Log912

9. x = 3

10.. x = 2 Log43 + 1

11. x = ±sqrt(Ln(100)

12.  x = -Ln(2)

13. x = 2

14. x = sqrt(3)

15. x = ± sqrt(2)^y - 6

16. x = 1 or x = 100