Transcript Angles, Degrees, and Special Triangles

Exponential & Logarithmic
Equations
MATH 109 - Precalculus
S. Rook
Overview
• Section 3.4 in the textbook:
– Solving exponential equations
– Solving logarithmic equations
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Solving Exponential Equations
Solving Simple Exponential
Equations
• Exponential Equation: an equation where the
variable to be solved for appears in the
exponent
• Recall the one-to-one property for exponents:
– If ax = ay, then x = y provided that a > 0 and a ≠ 1
– Provided that we can write both sides with the
same base a
• Often easier to write fractions as an integer with a
negative exponent
x
1
1
e.g.  
1 x
x ,
4x 
 4 x  4 3
 
   2
2
2
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Solving Simple Exponential
Equations (Example)
Ex 1: Solve the exponential equation:
x
a)
4  16
x
b)
1
   32
2
x
c)  3   16
9
4
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Solving More Complicated Exponential
Equations
• If we CANNOT write both sides with the same base,
then we need to get the variable out of the exponent
• Recall that a logarithm with base a is the inverse of
an exponential with base a
– i.e.
loga a x  x
• Thus, by introducing loga into the exponential
equation, the variable to be solved for is removed
from the exponent
– Think of loga as an operator which you MUST apply to
both sides of the equation
e.g. 6x  2  log6 6x  log6 2  x  log6 2
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Solving More Complicated Exponential
Equations (Continued)
• Before applying loga we must isolate the
exponential function ax
e.g. 1  2  3x  9  2  3x  8  3x  4
• Two forms to express the solution:
– Exact answers leave the log
– Approximate answers are converted into decimals
• Wait until the very end to evaluate the log
– This way there will be less error in the final answer
– More than likely will need to use the change-ofbase formula
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Solving More Complicated
Exponential Equations (Example)
Ex 2: Solve the exponential equation: i) exactly
ii) approximately to 3 decimal places
a) 43   20
x
c) e  4e  5
2x
x


3 x 1
6
2
7  9
b)
d)
500
 20
x/2
100  e
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Solving More Complicated
Exponential Equations (Example)
Ex 3: Solve and approximate to three decimal
places:
 0.065
1 

365 

365t
4
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Solving Logarithmic Equations
Solving Simple Logarithmic
Equations
• Logarithmic Equation: an equation where
logarithms appear
• Recall the one-to-one property for
logarithms:
logau = logav → u = v
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Solving Simple Logarithmic
Equations (Example)
Ex 4: Solve the logarithmic equation:
log17 3x  2  log17 9
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Solving More Complicated
Logarithmic Equations
• If we cannot apply the one-to-one property for
logarithms, we need to eliminate the logarithm
• Recall that an exponential with base a is the inverse
of a logarithm with base a
– i.e.
a loga x  x
• Thus, by introducing the exponential base a into the
logarithmic equation, the variable to be solved for is
removed from the logarithm
– Think of the exponential base a as an operator which
you MUST apply to both sides of the equation
• e.g. log4 x  3  4log4 x  43  x  64
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Solving More Complicated
Logarithmic Equations (Continued)
• Before we apply the exponential base of a, we need
to ensure that there is at most 1 logarithm on each
side of the equation
– Use the logarithm properties to combine logarithms
of the same base on each side
• Be aware of extraneous solutions
– Recall that the domain of the logarithm logax is x > 0
– Thus, if the argument to the logarithm is ≤ 0 when
checking a potential solution, the solution is
extraneous and we must discard it
– ALWAYS check the solutions to a logarithmic
equation!
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Solving More Complicated
Logarithmic Equations (Example)
Ex 5: Solve the logarithmic equation:
b) ln x  lnx  1  2
a) ln x  3
c) log2 x  log2 x  2  log2 x  6


d) log8x  log 1  x  2
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Summary
• After studying these slides, you should be able to:
– Solve exponential equations
– Write either an exact answer or an approximate answer
when appropriate
– Solve logarithmic equations and be able to identify
possible extraneous solutions
• Additional Practice
– See the list of suggested problems for 3.4
• Next lesson
– Exponential & Logarithmic Models (Section 3.5)
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