Pc Solve Exp And Log Eq

Solve E xpon en tial Equa tio ns

MATH PRO JEC T

SOLVING EXPONENTIAL EQUATIONS

If two powers with the same base are equal, then their exponents must be equal. For b > 0 and b ≠ 1, if b x = b y, then x = y.

Solving by Equating Exponents

SOLUTION

4 3x4=3x8=x +81x + 1. Solve

Write original equation.

Rewrite each power with base 2 . CHECK Check the solution by substituting it into the original equation. Power of a power property 22 (3x) = 23(x + 1) 3•1 1+1 Solve for x. 4 = 8 6x 3x + 3

( 22)3x = ( 23) x + 1

2 =2

Solution checks. 64 = 64 Equate exponents. 6x = 3x + 3

x=1 The solution is 1.

Solve for x.

Solving by Equating Exponents

When it is not convenient to write each side of an exponential equation using the same base, you can solve the equation by taking a logarithm of each side.

Taking a Logarithm of Each Side 2x – 3

Solve 10 S OLUTION

+ 4 = 21.

10 2 x – 3 + 4 = 21 10 2 x – 3 = 17 log 10 2 x – 3 = log 17 2 x – 3 = log 17 2 x = 3 + log 17

Write original equation. Subtract 4 from each side. Take common log of each side.

log 10 x = x  

Add 3 to each side.

x = 1 (3 + log 17 ) 2

1 Multiply each side by      . 

x ≈ 2.115

Use a calculator.

2

Taking a Logarithm of Each Side

Solve 10 2 x – 3 + 4 = 21. CHECK

Check the solution algebraically by substituting into the original equation. Or, check it graphically by graphing both sides of the equation and observing that the two graphs intersect at x ≈ 2.115. y

y= 21

y = 10 2 x – 3 + 4

1.0

2.0

x

SOLVING LOGARITHMIC EQUATIONS

To solve a logarithmic equation, use this property for logarithms with the same base: For positive numbers b, x, and y where b ≠ 1,

log b x = log b y if and only if x = y.

Solving a Logarithmic Equation

Solve log 3 (5 x – 1) = log 3 (x + 7) . SOLUTION CHECK

Check the solution by substituting it into the original equation.

log 3 (5 x – 1) = log 3 (x + 7)

Write original equation.

log 3 (5 x – 1) = log 3 (x + 7) 5x – 1 = x + 7 ? log 3 (5 · 2 – 1) = log 3 (2 + 7) 5x = x + 8

Write original equation. Use property for logarithms  with the same base. Substitute 2 for x. Add 1 to each side.

log 3 9 = log 3 9 x=2 The solution is 2.

Solution checks. Solve for x.

Solving a Logarithmic Equation

Solve log 5 (3x + 1) = 2 . SOLUTION CHECK

Check the solution by substituting it into the original equation.

log 5 (3x + 1) = 2 log 5 (3x + 1) = 2 log (3x + 1) 5 5 = 5?2 log 5 (3 · 8 + 1) = 2 ? 3x + 1 = 25 log 5 25 = 2

x =28= 2 The solution is 8.

Write original equation. Write original equation.

Exponentiate each side using base 5. Substitute 8 for x. log b          = x Simplify. b x  

Solution checks. Solve for x.

Checking for Extraneous Solutions

Because the domain of a logarithmic function generally does not include all real numbers, you should be sure to check for extraneous solutions of logarithmic equations. You can do this algebraically or graphically.

Checking for Extraneous Solutions

SOLUTION

log 5 xlog + log – 1)(x= –2 1) = 2Write original equation. Solve 5 x (x + log . Check for extraneous solutions. Product property of logarithms. log [ 5 x (x – 1)] = 2 10

log (5 x 2 – 5 x)

= 10 2 Exponentiate each side using  base 10.

5 x 2 – 5 x = 100 x 2 – x – 20 = 0 (x – 5 )(x + 4) = 0 x = 5 or x = – 4

10 log x = x  

 

Write in standard form. Factor. Zero product property

Checking for Extraneous Solutions

SOLUTION

log 5 x + log (x – 1) = 2 x = 5 or x = – 4

Check for extraneous solutions. Zero product property

The solutions appear to be 5 and – 4. However, when you check these in the original equation or use a graphic check as shown below, you can see that x = 5 is the only solution. y

y=2 x

y = log 5 x + log (x – 1)

Using a Logarithmic Model

SEISMOLOGY On

May 22, 1960, a powerful earthquake took place in Chile. It had a moment magnitude of 9.5. How much energy did this earthquake release?

The moment magnitude M of an earthquake that releases energy E (in ergs) can be modeled by this equation:

M = 0.291 ln E + 1.17

Using a Logarithmic Model

SOLUTION

M = 0.291 ln E + 1.17 Write model for moment magnitude. 9.5 = 0.291 ln E + 1.17 Substitute 9.5 for M.  8.33 = 0.291 ln E

Subtract 1.17 from each side. 

28.625 ≈ ln E

Divide each side by 0.291.

e 28.625 ≈ e ln E

Exponentiate each side using base e.

2.702 x 1012 ≈ E

e ln x = e log e x = x  

The earthquake released about 2.7 trillion ergs of energy.

SOLVING LOGARITHMIC EQUATIONS EXPONENTIAL AND LOGARITHMIC PROPERTIES x

y

1

For b > 0 and b ≠ 1, if b  = b   , then x = y.

2

For positive numbers  b, x, and y  where b ≠  1,

 

 

log b x = log b y  if and only if  x = y.  

3

 

x

y

For b > 0 and b ≠  1, if x = y, then b  = b   .