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# Simple Exponent and Logarithm Equations Help

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## Introduction to Simple Exponent and Logarithm Equations

Equations with exponents and logarithms come in many forms. Sometimes more than one strategy will work to solve them. We will first solve equations of the form “log = number” and “log = log.” We will solve an equation of the form “log = number” by rewriting the equation as an exponential equation.

#### Examples

Solve the equation for x .

• log 3 ( x + 1) = 4
• Rewrite the equation as an exponential equation.

log 3 ( x + 1) = 4

3 4 = x + 1

81 = x + 1

80 = x

• log 2 (3 x − 4) = 5
• 2 5 = 3 x − 4

32 = 3 x − 4

12 = x

### Cancellation Law

The logarithms cancel for equations in the form “log = log” as long as the bases are the same. For example, the solution to the equation log 8 x = log 8 10 is x = 10. The cancelation law a log a x = x makes this work.

log 8 x = log 8 10

8 log 8 x = 8 log 8 10

x = 10 (By the cancelation law)

#### Examples

Solve for x .

• log 6 ( x + 1) = log 6 2 x
• log 6 ( x + 1) = log 6 2 x

x + 1 = 2 x The logs cancel.

1 = x

• log 4 = log( x − 1)

log 4 = log( x − 1)

4 = x − 1 The logs cancel.

5 = x

### Simple Exponent and Logarithm Equations Practice Problems

#### Practice

Solve for x .

1. log 7 (2 x + 1) = 2
2. log 4 ( x + 6) = 2
3. log 5 x = 1
4. log 2 (8 x − 1) = 4
5. log 3 (4 x − 1) = log 3 2
6. log 2 (3 − x ) = log 2 17
7. ln 15 x = In( x + 4)

#### Solutions

1. log 7 (2 x + 1) = 2

7 2 = 2 x + 1

24 = x

2. log 4 ( x + 6) = 2

4 2 = x + 6

10 = x

3. log 5 x = 1

10 1 = 5 x

2 = x

4.

5.

6. log 2 (3 − x ) = log 2 17

3 − x = 17

x = −14

7.

8.

## Finding Approximate Solutions - Converting Exponential Equations to Logarithmic Equations

We need to use calculators to find approximate solutions for exponential equations whose base is e or 10. We will rewrite the exponential equation as a logarithmic equation, solve for x , and then use a calculator to get an approximate solution.

#### Examples

Solve for x. Give solutions accurate to four decimal places.

• e 2 x = 3
• 10 x +1 = 9
• 2500 = 1000 e x −4

### Finding Approximate Solutions Practice Problems

#### Practice

Solve for x . Give your solutions accurate to four decimal places.

1. 10 3 x = 7
2. e 2 x +5 = 15
3. 5000 = 2500 e 4 x
4. 32 = 8 · 10 6 x −4
5. 200 = 400 e −0.06 x

#### Solutions

1. 10 3 x = 7

1. e 2 x +5 = 15

2. 5000 = 2500 e 4 x

3. 32 = 8 · 10 6 x −4 Divide both sides by 8.

4. 200 = 400 e −0.06 x

## Graphing Simple Exponent and Logarithm Equations

The logarithm function f ( x ) = log a x is the inverse of g ( x ) = a x. The graph of f ( x ) is the graph of g ( x ) with the x - and y-values reversed. To sketch the graph by hand, we will rewrite the logarithm function as an exponent equation and graph the exponent equation.

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