Expand the following logLog3(x2yz / w9)

Condense the following logLog 2(x) + log2(y) - 3log2(z)

Essential Question: How do you solve exponential and logarithmic equations?

Standard: MM4A4. Students will investigate functions.

a. Compare and contrast properties of functions within and across the following types: linear, quadratic, polynomial, power, rational, exponential, logarithmic, trigonometric, and piecewise.

Exponential Equations

Exponential Equations

An exponential equation is one in

which the variable occurs in the

exponent.

• For example, 2x = 7

Solving Exponential Equations

The variable x presents a difficulty

because it is in the exponent.

• To deal with this difficulty, we take the logarithm of each side and then use the Laws of Logarithms to “bring down x” from the exponent.

Solving Exponential Equations

• The power rulesays that: logaAC = C logaA

(Law 3)

2 7

ln2 ln7

ln2 ln7

ln72.807

ln2

x

x

x

x

Guidelines for Solving Exponential Equations

1. Isolate the exponential expression on

one side of the equation.

2. Take the logarithm of each side,

and then use the Laws of Logarithms

to “bring down the exponent.”

3. Solve for the variable.

E.g. 1—Solving an Exponential Equation

Find the solution of

3x + 2 = 7

correct to six decimal places.

• We take the common logarithm of each side and use the power rule.

E.g. 1—Solving an Exponential Equation

2

2

(Law 3)

3 7

log 3 log7

2 log3 log7

log72

log3

log72 0.228756

log3

x

x

x

x

x

E.g. 2—Solving an Exponential Equation

Solve the equation 8e2x = 20.

• We first divide by 8 to isolate the exponential term on one side.

2

2 208

2

8 20

ln ln2.5

2 ln2.5

ln2.50.458

2

x

x

x

e

e

e

x

x

E.g. 3—Solving Algebraically and Graphically

Solve the equation

e3 – 2x = 4

algebraically and graphically

E.g. 3—Solving Algebraically

The base of the exponential term is e.

So, we use natural logarithms to solve.

• You should check that this satisfies the original equation.

Solution 1

3 2

3 2

12

4

ln ln4

3 2 ln4

2 3 ln4

3 ln4 0.807

x

x

e

e

x

x

x

E.g. 4—Exponential Equation of Quadratic Type

Solve the equation e2x – ex – 6 = 0.

• To isolate the exponential term, we factor.

2

2(Law of Exponents)

(Zero-Product Property)

6 0

6 0

3 2 0

3 0 or 2 0

3 2

x x

x x

x x

x x

x x

e e

e e

e e

e e

e e

E.g. 4—Exponential Equation of Quadratic Type

The equation ex = 3 leads to x = ln 3.

However, the equation ex = –2 has no solution

because ex > 0 for all x.

• Thus, x = ln 3 ≈ 1.0986 is the only solution.

• You should check that this satisfies the original equation.

Logarithmic Equations

Logarithmic Equations

A logarithmic equation is one in which

a logarithm of the variable occurs.

• For example,

log2(x + 2) = 5

Solving Logarithmic Equations

To solve for x, we write the equation

in exponential form.

x + 2 = 25

x = 32 – 2

= 30

Solving Logarithmic Equations

Another way of looking at the first step

is to raise the base, 2, to each side.

2log2(x + 2) = 25

x + 2 = 25

x = 32 – 2 = 30

• The method used to solve this simple problem is typical.

Guidelines for Solving Logarithmic Equations

1. Isolate the logarithmic term on one side

of the equation.• You may first need to combine the logarithmic

terms.

2. Write the equation in exponential form

(or raise the base to each side).

3. Solve for the variable.

E.g. 6—Solving Logarithmic Equations

Solve each equation for x.

(a) ln x = 8

(b) log2(25 – x) = 3

E.g. 6—Solving Logarithmic Eqns.

ln x = 8

x = e8

Therefore, x = e8 ≈ 2981.

• We can also solve this problem another way:

Example (a)

ln 8

8

ln 8x

x

e e

x e

E.g. 6—Solving Logarithmic Eqns.

The first step is to rewrite the equation

in exponential form.

2

3

log 25 3

25 2

25 8

25 8

17

x

x

x

x

Example (b)

E.g. 7—Solving a Logarithmic Equation

Solve the equation

4 + 3 log(2x) = 16

• We first isolate the logarithmic term.

• This allows us to write the equation in exponential form.

E.g. 7—Solving a Logarithmic Equation

4

4 3 log 2 16

3 log 2 12

log 2 4

2 10

5000

x

x

x

x

x

Examples: Solve the following: ex = 32 3(2)x = 42

More examplesSolve 4e2x -3 = 2

2x = 512

Solve the following

Ln(5) – ln(x) = 0

Ln(x) = -8

2(32t -5 ) -4 = 11

e2x -3ex + 2 = 0

Ln(3x) = 2

Log3 (5x-1) = Log3(x+7)

Solve

•(1/2)x = 32

P 221 # 1-8, 17, 22, 27, 29-34, 39-44,85,96, 91,92