3.3 Logarithmic Functions

OBJECTIVES
1. Converting from Exponential Expressions ← → Logarithmic Expressions
2. Evaluate Logarithmic Expressions
3. Graphing Logarithmic Functions
4. Transformations on Logarithmic Functions
5. Solving Logarithmic Functions

Objective 1: Exponential Expressions ← → Logarithmic Expressions

The Logarithmic Function
The logarithmic function to the base “a”, when a>0 and “a” is not equal to 1, is denoted as y=logax (read as “y is the logarithm to the base a of x”) and is defined by

y=logax if and only if x=ay

Relating Logarithms to Exponents:

(a) If y=log5x, then x=5y. For example, 2=log525 is equivalent to 25=52

(b) If y=log8x, then x=8y. For example, -1=log8(⅛) is equivalent to ⅛= 8-1

Exponential→ Logarithmic Expressions:
Change the following expressions to an equivalent expression involving a logarithm

(a) e2=12

(b) 2.47=l5

(c) b5=90

Use the property y=logax and x=ay, a>0, a≠1, are equivalent

(a) If e2=12, then 2=loge12

(b) If 2.47=n, then 7=log2.4n

(c) If b5=90, then 5=logb90

Logarithmic→ Exponential Expressions:
Change the following expressions to an equivalent expression involving an exponent

(a) logm8=24 → m24=8

(b) logen=-9→ e-9=n

(c) log720=h→ 7h=20

Objective 2: Evaluate Logarithmic Expressions

Finding the Exact Value of a Logarithmic Expression:

Find the exact value of:

(a) log636

(b) log4(1/64)

Solution to (a):
y=log636
6y=36 *change to exponential form 36=62
6y=62
y=2
Therefore, log636=2

Solution to (b):
y=log4(1/64)
4y= 1/64 *change to exponential form 1/64=1/(4-3)
4y= 1/(4-3)
y=-3
Therefore, log4(1/64)=-3

Objective 3: Graphing Logarithmic Functions

Exponential and logarithmic functions are inverse functions of each other. The logarithmic function, y=logax, is a reflection of the exponential function, y=ax, over y=x.

log005.gif

Standard form of a logarithmic function is y=logax

  • Restrictions on “a”
    • a>0
      • Reasoning: It is impossible to take even roots of negative numbers
    • a≠1
      • Reasoning:Any number to the first power is still one
  • Restrictions on “x”
    • x>0
      • Reasoning: “x” has to be positive

Properties of the Logarithmic Function f(x)=logax

1. Domain: (0, ∞)

2. Range: (-∞,∞)

3. The x-intercept is (1,0), there is no y-intercept

4. The y-axis (x=0) is a vertical asymptote

5. The graph of the function contains the points (1,0), (a,1), and (1/a,-1)

6. The graph is continuous and is a smooth curve (no corners or gaps)

Graph of f(x)=logax
log-function.gif

The natural logarithm is given the special symbol, ln, because it occurs often in applications. The base of the natural log is always e. The inverse of y=ln(x) is x=ey.

logs.h4.gif

The common logarithmic function has base 10. The notation for this would be y=log(x) if and only if x=10y. If a logarithmic function doesn’t have a base “a” indicated, it is assumed to be 10. The function y=log(x) and x=10y are inverse functions.

MG12C6_008.png

Objective 4: Transformations of Logarithmic Functions

Natural Log Transformations
The parents function of the natural log is f(x)=ln(x). When the function is not in standard form, transformations are used to graph the function:

  • If f(x)=-ln(x) → reflection over the x-axis
  • If f(x)=ln(-x) → reflection over the y-axis
  • If f(x)=ln(x+b) → horizontal shift
    • If “b” is negative (-), then horizontal shift right “b” units
    • If “b” is positive (+), then horizontal shift “b” units left
    • If there is a horizontal shift, the horizontal asymptote of the functions is x=b
  • If f(x)=ln(x)+b → vertical shift
    • If “b” is negative (-), vertical shift down “b” units
    • If “b” is positive (+), vertical shift up “b” units

Example:
List the transformation on the function f(x)=-ln(x-2)+5
1. Reflection over the x-axis
2. Vertical shift up 5 units
3. Horizontal shift 2 units right

Common Log Transformations
The parents function of the common log is f(x)=log(x). When the function is not in standard form, transformations are used to graph the function:

  • If f(x)=log(-x) → reflection over the y-axis
  • If f(x)=log(x+b) → horizontal shift
    • If “b” is negative (-), then horizontal shift right “b” units
    • If “b” is positive (+), then horizontal shift “b” units left
    • If there is a horizontal shift, the horizontal asymptote of the functions is x=b
  • If f(x)=log(x)+b → vertical shift
    • If “b” is negative (-), vertical shift down “b” units
    • If “b” is positive (+), vertical shift up “b” units

Example:
List the transformations on the function f(x)=log(-x+5)-9
1. Reflection over the y-axis
2. Horizontal shift 5 units left
3. Vertical shift 9 units down

Objective 5: Solving Logarithmic Functions
Logarithmic equations are equations that contain logarithms. Logarithmic equations can be solved by changing from logarithmic form to exponential form.

Solving Logarithmic Equations

Examples:
Solve: log3(1/9)=x
3x=1/9 *change to exponential form*
x=-2

Solve: log3N=2.1
32.1=N *change to exponential form*
N=10.045

Solve: logx36=2
x2=36 *change to exponential form*
x=6

Solve: log4(2x-4)=2
42=2x-4 *change to exponential form*
16=2x-4
20=2x
x=10

Check: log4(2(10)-4)=2
log4(16)=2
42=16

Using Logarithms to solve Exponential functions
An exponential equation can be solved by changing it into logarithmic form

Examples:
Solve: e5x=8
5x=lne8 *natural log (ln) is always base e
x=(ln8)/(5)
X≈.41588830

Practice Problems:
For Problems 1-5, change each exponential expression to the equivalent logarithmic expression:
1. 25 = 52
2. 16 = 42
3. ex = 4
4. 2.73 = x
5. 5z = 5

For Problems 6-10, change each logarithmic expression to the equivalent exponential expression:
6. log56=z
7. ln7=x
8. log3w=4.1
9. log5x=9.7
10. lnx=5

For Problems 11-15, find the exact value without using a calculator:
11. log525
12. log1010
13. log3(1/27)
14. lne3
15. log1/39