example 2 take a logarithm of each side solve 4 = 11. x 4 = 11 x log 4 = log 11 x 4 4 log 4 x = 11 x...

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EXAMPLE 2 Take a logarithm of each side Solve 4 = 11. x 4 = 11 x log 4 = log 11 x 4 4 log 4 x = 11 x = log 11 log 4 x 1.73 SOLUTION Write original equation. Take log of each side. 4 Change-of-base formula Use a calculator. log b = x b x The solution is about 1.73. Check this in the original equation. ANSWER

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Page 1: EXAMPLE 2 Take a logarithm of each side Solve 4 = 11. x 4 = 11 x log 4 = log 11 x 4 4 log 4 x = 11 x = log 11 log 4 x 1.73 SOLUTION Write original equation

EXAMPLE 2 Take a logarithm of each side

Solve 4 = 11.x

4 = 11x

log 4 = log 11x

4 4

log4x = 11

x = log 11 log 4

x 1.73

SOLUTION

Write original equation.

Take log of each side.4

Change-of-base formula

Use a calculator.

log b = xbx

The solution is about 1.73. Check this in the original equation.

ANSWER

Page 2: EXAMPLE 2 Take a logarithm of each side Solve 4 = 11. x 4 = 11 x log 4 = log 11 x 4 4 log 4 x = 11 x = log 11 log 4 x 1.73 SOLUTION Write original equation

EXAMPLE 3 Use an exponential model

You are driving on a hot day when your car overheats and stops running. It overheats at 280°F and can be driven again at 230°F. If r = 0.0048 and it is 80°F outside,how long (in minutes) do you have to wait until you can continue driving?

Cars

Page 3: EXAMPLE 2 Take a logarithm of each side Solve 4 = 11. x 4 = 11 x log 4 = log 11 x 4 4 log 4 x = 11 x = log 11 log 4 x 1.73 SOLUTION Write original equation

EXAMPLE 3 Use an exponential model

SOLUTION

T = ( T – T )e + T° R– rt

R

230 = (280 – 80)e + 80–0.0048t

150 = 200e–0.0048t

0.75 = e–0.0048t

ln 0.75 = ln e –0.0048t

– 0.2877 – 0.0048t

60 t

Newton’s law of cooling

Subtract 80 from each side.

Divide each side by 200.

Take natural log of each side.

Divide each side by – 0.0048.

Substitute for T, T , T , and r.° R

In e = log e = xexx

Page 4: EXAMPLE 2 Take a logarithm of each side Solve 4 = 11. x 4 = 11 x log 4 = log 11 x 4 4 log 4 x = 11 x = log 11 log 4 x 1.73 SOLUTION Write original equation

EXAMPLE 3 Use an exponential model

You have to wait about 60 minutes until you can continue driving.

ANSWER

Page 5: EXAMPLE 2 Take a logarithm of each side Solve 4 = 11. x 4 = 11 x log 4 = log 11 x 4 4 log 4 x = 11 x = log 11 log 4 x 1.73 SOLUTION Write original equation

GUIDED PRACTICE for Examples 2 and 3

Solve the equation.

4. 2 = 5x

Log 2 = log 5x

2 2

log2x = 5

Write original equation.

Take log of each side.2

log b = xbx

SOLUTION

2 = 5x

x = log 5 log 2 Change-of-base formula

x 2.32 Use a calculator.

Page 6: EXAMPLE 2 Take a logarithm of each side Solve 4 = 11. x 4 = 11 x log 4 = log 11 x 4 4 log 4 x = 11 x = log 11 log 4 x 1.73 SOLUTION Write original equation

GUIDED PRACTICE for Examples 2 and 3

The solution is about 2.32. Check this in the original equation.

ANSWER

Page 7: EXAMPLE 2 Take a logarithm of each side Solve 4 = 11. x 4 = 11 x log 4 = log 11 x 4 4 log 4 x = 11 x = log 11 log 4 x 1.73 SOLUTION Write original equation

GUIDED PRACTICE for Examples 2 and 3

Solve the equation.

5. 7 = 159x

Log 7 = log 159x

7 7

9x = 15 log7

Write original equation.

Take log of each side.7

log b = xbx

SOLUTION

9x = log 15 log 7 Change-of-base formula

x 0.155 Use a calculator.

7 = 159x

Page 8: EXAMPLE 2 Take a logarithm of each side Solve 4 = 11. x 4 = 11 x log 4 = log 11 x 4 4 log 4 x = 11 x = log 11 log 4 x 1.73 SOLUTION Write original equation

GUIDED PRACTICE for Examples 2 and 3

The solution is about 0.155. Check this in the original equation.

ANSWER

Page 9: EXAMPLE 2 Take a logarithm of each side Solve 4 = 11. x 4 = 11 x log 4 = log 11 x 4 4 log 4 x = 11 x = log 11 log 4 x 1.73 SOLUTION Write original equation

GUIDED PRACTICE for Examples 2 and 3

Solve the equation.

6. 4e –7 = 13 –0.3x

SOLUTION

4e –7 = 13 –0.3x

4e = 13 + 7 –0.3x

4e = 20 –0.3x

e = = 5 –0.3x 20 4

log e = log 5–0.3xe e

Page 10: EXAMPLE 2 Take a logarithm of each side Solve 4 = 11. x 4 = 11 x log 4 = log 11 x 4 4 log 4 x = 11 x = log 11 log 4 x 1.73 SOLUTION Write original equation

GUIDED PRACTICE for Examples 2 and 3

The solution is about 5.365. Check this in the original equation.

ANSWER

– 0.3x = 1.6094

x = = – 5.365– 1.6094 0.3

– 0.3x = In 5 In e = log e = xexx

Divide each side by – 0.3x.

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