**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=log_{a}x (read as “y is the logarithm to the base a of x”) and is defined by

**y=log _{a}x if and only if x=a^{y}**

**Relating Logarithms to Exponents:**

**(a)** If y=log_{5}x, then x=5^{y}. For example, 2=log_{5}25 is equivalent to 25=5^{2}

**(b)** If y=log_{8}x, then x=8^{y}. For example, -1=log_{8}(⅛) is equivalent to ⅛= 8^{-1}

**Exponential→ Logarithmic Expressions:**

Change the following expressions to an equivalent expression involving a logarithm

**(a)** e^{2}=12

**(b)** 2.4^{7}=l5

**(c)** b^{5}=90

Use the property y=log_{a}x and x=a^{y}, a>0, a≠1, are equivalent

**(a)** If e^{2}=12, then 2=log_{e}12

**(b)** If 2.4^{7}=n, then 7=log_{2.4}n

**(c)** If b^{5}=90, then 5=log_{b}90

**Logarithmic→ Exponential Expressions:**

Change the following expressions to an equivalent expression involving an exponent

**(a)** log_{m}8=24 → m^{24}=8

**(b)** log_{e}n=-9→ e^{-9}=n

**(c)** log_{7}20=h→ 7^{h}=20

**Objective 2:** Evaluate Logarithmic Expressions

**Finding the Exact Value of a Logarithmic Expression:**

Find the exact value of:

**(a)** log_{6}36

**(b)** log_{4}(1/64)

**Solution to (a):**

y=log_{6}36

6^{y}=36 *change to exponential form 36=6^{2}

6^{y}=6^{2}

y=2

Therefore, log_{6}36=2

**Solution to (b):**

y=log_{4}(1/64)

4^{y}= 1/64 *change to exponential form 1/64=1/(4^{-3})

4^{y}= 1/(4^{-3})

y=-3

Therefore, log_{4}(1/64)=-3

**Objective 3:** Graphing Logarithmic Functions

Exponential and logarithmic functions are inverse functions of each other. The logarithmic function, y=log_{a}x, is a reflection of the exponential function, y=a^{x}, over y=x.

Standard form of a logarithmic function is y=log_{a}x

- 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

- a>0
- Restrictions on “x”
- x>0
- Reasoning: “x” has to be positive

- x>0

**Properties of the Logarithmic Function f(x)=log _{a}x**

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)=log

_{a}x

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=e^{y}.

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

**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: log_{3}(1/9)=x

3^{x}=1/9 *change to exponential form*

x=-2

Solve: log_{3}N=2.1

3^{2.1}=N *change to exponential form*

N=10.045

Solve: log_{x}36=2

x^{2}=36 *change to exponential form*

x=6

Solve: log_{4}(2x-4)=2

4^{2}=2x-4 *change to exponential form*

16=2x-4

20=2x

x=10

Check: log_{4}(2(10)-4)=2

log_{4}(16)=2

4^{2}=16

**Using Logarithms to solve Exponential functions**

An exponential equation can be solved by changing it into logarithmic form

**Examples:**

Solve: e^{5x}=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 = 5^{2}

2. 16 = 4^{2}

3. e^{x} = 4

4. 2.7^{3} = x

5. 5^{z} = 5

For Problems 6-10, change each logarithmic expression to the equivalent exponential expression:

6. log_{5}6=z

7. ln7=x

8. log_{3}w=4.1

9. log_{5}x=9.7

10. lnx=5

For Problems 11-15, find the exact value without using a calculator:

11. log_{5}25

12. log_{10}10

13. log_{3}(1/27)

14. lne_{3}

15. log_{1/3}9