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6.6 Logarithmic and Exponential Equations

In this section, we will study the following topics:

Solving logarithmic equations

Solving exponential equations

Using exponential and logarithmic equations to solve real-life problems.

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Strategies for Solving Logarithmic Equations

There are two basic strategies for solving logarithmic

equations—

1. Converting the log equation into an exponential

equation by using the definition of a logarithm:

2. Using the One-to-One Property:

0; 1log ya a awherex y iff a x

0;where , are positve; 1

If log log then

M N a

a a

a

M N M N

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Solving Logarithmic Equations

I. Converting to Exponential Form

1. ISOLATE the logarithmic expression on one side

of the equation.

2. CONVERT TO EXPONENTIAL FORM

3. SOLVE for x. Give approximate answers to 3 decimal places,

unless otherwise indicated.

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Example: Converting to Exponential Form

Solve 2ln( 5) 7 13x

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Solving Logarithmic Equations

II. Using the One-to-One* Properties

*Can be used if the equation can be written so that both sides are expressed as SINGLE LOGARITHMS with the SAME BASE.

1. Use the properties of logarithms to CONDENSE the log expressions on either side of the equation into SINGLE LOG expressions.

2. Apply the ONE-TO-ONE PROPERTY.

3. SOLVE for x. Give approximate answers to 3 decimal places, unless otherwise indicated.

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Example: One-to-One Property for Logs

Solve log3x + 2log35 = log3(x + 8)

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41) 4 7log x

PRACTICE!!

Solve each of the following LOGARITHMIC equations.

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23 3 32) log log 8 log 8x x x

PRACTICE!!

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PRACTICE!!

3) 3ln 2 1.5x

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4) ln ln 2 ln5x

PRACTICE!!

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Strategies for Solving Exponential Equations

There are two basic strategies for solving exponential

equations—

1. Using the One-to-One Property:

2. Taking the natural or common log of each side.

0;where 1If then au v

aa a u v

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Solving Exponential Equations

I. Using the One-to-One property

*Can be used if the equation can be written so that both sides are expressed as powers of the SAME BASE

Use the properties of exponents to CONDENSE the exponential expressions on either side of the equation into SINGLE exponential expressions.

Apply the ONE-TO-ONE PROPERTY.

SOLVE for x. Give approximate answers to 3 decimal places, unless otherwise indicated.

0;where 1If then au v

aa a u v

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Example: One-to-One Property

Solve 25 4 59x

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Example: One-to-One Property

Solve5 12 8 32x x x

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II. Taking the Natural or Common Log of Each Side

If there is one exponential term, ISOLATE the

exponential term on one side of the equation.

TAKE THE NATURAL OR COMMON LOG OF EACH

SIDE of the equation.

Use the property to get the variable out

of the exponent.

SOLVE for x. Give approximate answers to 3 decimal places, unless

otherwise indicated.

Solving Exponential Equations

log logrM r M

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Solve 3(54x+1) -7 = 10

Example: Taking the Log of Each Side

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Solve.

Example: Taking the Log of Each Side

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53

xx

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Solve each of the following EXPONENTIAL equations.

11) 7

49x 2) 3 10 570x

PRACTICE!!

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PRACTICE!!

23) 5 6 0x xe e

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5 34) 6 2x x

PRACTICE!!

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Solving Exponential and Logarithmic Equations Graphically

Remember, you can verify the solution of any one of these equations by finding the graphical solution using your TI-83/84 calculator.

Enter the left hand side of the original equation in y1

Enter the right side in y2

Find the point at which the graphs intersect.

Below is the graphical solution of the equation 2ln(x - 5) = 6

The x-coordinate of the intersection point is approximately 25.086. This is the (approx) solution of the equation.

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The population of Asymptopia was 6500 in 1970 and has been tripling every

12 years since then. In what year will the population reach 75,000?

Let t represent the number of years since 1970; P(t) represents the population after t years.

12( ) 6500 3t

P t

Exponential Growth Example

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A Compound Interest Example

How long will it take $25,000 to grow to $500,000 at 9% annual interest compounded continuously?

Use the compound interest formula: r tA Pe

Where P = Principal (original amount invested or borrowed)

r = annual interest rate

t = number of years money is invested

A = the amount of the investment after t years

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A Compound Interest Example (cont.)

Substitute in the given values and solve for t.

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End of Section 6.6