If two powers with the same base are equal, then their exponents must be equal'

1
If two powers with the same base are equal,then
their exponents must be equal.
2
Solve 43x 8 x 1.
4 3x 8 x 1
SOLUTION
Write original equation.
( 22)3x ( 23) x 1
Rewrite each power with base 2 .
Check the solution by substituting it into the
original equation.
22 (3x) 23(x 1)
Power of a power property
4 3 1 8 1 1
Solve for x.
26x 23x 3
64 64
Solution checks.
6x 3x 3
Equate exponents.
x 1
Solve for x.
The solution is 1.
3
When it is not convenient to write each side of
an exponential equation using the same base, you
cansolve the equation by taking a logarithm of
each side.
4
Solve 10 2 x 3 4 21.
SOLUTION
10 2 x 3 4 21
Write original equation.
10 2 x 3 17
Subtract 4 from each side.
log 10 2 x 3 log 17
Take common log of each side.
2 x 3 log 17
log 10 x x
2 x 3 log 17
Add 3 to each side.
x ? 2.115
Use a calculator.
5
Solve 10 2 x 3 4 21.
Check the solution algebraically by substituting
into theoriginal equation. Or, check it
graphically by graphingboth sides of the
equation and observing that the two graphs
intersect at x ? 2.115.
y 10 2 x 3 4
y 21
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SOLVING LOGARITHMIC EQUATIONS
To solve a logarithmic equation, use
thisproperty 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.
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Solve log 3 (5 x 1) log 3 (x 7) .
SOLUTION
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)
Write original equation.
5 x 1 x 7
Use property for logarithms with the same base.
Substitute 2 for x.
5 x x 8
Add 1 to each side.
log 3 9 log 3 9
Solution checks.
x 2
Solve for x.
The solution is 2.
8
Solve log 5 (3x 1) 2 .
SOLUTION
Check the solution by substituting it into the
original equation.
log 5 (3x 1) 2
Write original equation.
log 5 (3x 1) 2
Write original equation.
Exponentiate each side using base 5.
Substitute 8 for x.
3x 1 25
Simplify.
2 2
Solution checks.
x 8
Solve for x.
The solution is 8.
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Or, you could do it another way
Solve log 5 (3x 1) 2 .

• Rewrite the equation in exponential form and
  solve it that way.

52 3x 1
25 3x 1
24 3x
8 x
10
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.
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Check for extraneoussolutions.
log 5 x log (x 1) 2
Solve log 5 x log (x 1) 2 .
SOLUTION
Write original equation.
log 5 x (x 1) 2
Product property of logarithms.
Exponentiate each side using base 10.
5 x 2 5 x 100
10 log x x
x 2 x 20 0
Write in standard form.
(x 5 )(x 4) 0
Factor.
x 5 or x 4
Zero product property
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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 2
y log 5 x log (x 1)
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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
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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.
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SOLVING LOGARITHMIC EQUATIONS
EXPONENTIAL AND LOGARITHMIC PROPERTIES
For b gt 0 and b ? 1, if b x b y , then x y.
For positive numbers b, x, and y where b ?
1, log b x log b y if and only if x y.
For b gt 0 and b ? 1, if x y, then b x b y .