Math 152 — Rodriguez
Blitzer — 9.5
Exponential and Logarithmic Equations
I. Solving Exponential Equations
A. An exponential equation is an equation containing the variable in the exponent.
B. Recall: An exponential function is one-to-one. This tells us: If bM = bN then M = N.
C. Solving exponential equations when both sides can be expressed with the SAME BASE
Steps:
1. Write both sides of the equation using the ________________.
2. Set the ___________ _____ to each other.
M=N
3. ____________ the equation.
bM = bN
Examples: Solve each exponential equation by expressing each side as a power of the
same base and then equating exponents.
1)
4)
7)
2
x+1
= 16
4 x = 32
42! x =
2 x+1
= 125
2) 5
5)
1
64
8)
9x =
1
3
3
125x = 625
3)
x +1
7 6
6)
3x!2 = 81
9) 25x =
= 7
1
5
C. Solving exponential equations when both sides CAN’T BE WRITTEN WITH THE
SAME BASE
Will use one of these:
Inverse Property
log1010M = M
Power Rule
logbMp = plogbM
lneM = M
Steps:
1. _________ the exponential expression on one side of the equation.
2. Take the ______________ of both sides. (Use base 10 or base e because these
are the buttons you have on your calc.)
3. Apply the inverse property or the power rule to get the variable ‘out of’ the
exponent.
4. _________ for the variable.
Examples: Solve each exponential equation by taking the logarithm on both sides.
Express the solution set in terms of logarithms. Then use a calculator to obtain a
decimal approximation, correct to two decimal places, for the solution.
1)
3)
2ex = 160
25e5 x = 950
5) 6e x = 240
2)
10 x = 81
4)
5 x! 3 = 137
6)
e1! x = 500
7) 2e3x = 126
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II. Solving Logarithmic Equations
A. A logarithmic equation is an equation containing a variable in the logarithmic
expression.
log ( x ! 3) + log x = 1
log 4 x = !3
log 6 ( x ! 3) + log 6 ( x + 3) = log 6 7
B. Useful Properties to Solve Logarithmic Equations − usual restrictions on b, M and N.
Product Rule:
Quotient Rule:
logbM + logbN = logb (MN)
"M%
logb M ! logb N = logb $ '
#N&
One-to-one property of logarithms: If logbM=logbN, then M=N.
C. Solving logarithmic equations
Steps:
1. Write the equation so that each side is ONE expression, either a number or a
logarithm. If necessary, use the Product or Quotient Rule to rewrite the
sum/difference of log’s as one logarithm. When you are done with this step your
equation must look like: logbM = N or logbM = logbN.
2. If the equation looks like:
logbM = N then rewrite the equation in exponential form
logbM = logbN then use the one-to-one property and set M=N
3. Solve the resulting equation.
4. Check the proposed solution(s) in the original equation. Make sure you are not
taking the logarithm of ___ or a _____________.
Examples: Solve each logarithmic equation. Be sure to reject any value of x that is not
in the domain of the original logarithmic expressions. Give the exact answer. Then,
where necessary use a calculator to obtain a decimal approximation, correct to two
decimal places, for the solution.
1) log 4 x = !3
2) log ( x ! 2 ) = 3
3) log 3 ( x + 4 ) ! log 3 ( x ! 4 ) = 2
4)
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8 + 3ln x = 20
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5)
log 4 ( x ! 5 ) = log 4 11
6) log 6 ( x ! 3) + log 6 ( x + 3) = log 6 7
7) log 5 ( x + 4 ) = 3
9) log ( x ! 3) + log x = 1
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8) 4 ln(3x) = 40
10)
log 2 x ! log 2 ( x ! 5 ) = log 2 4
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III.
Applications involving logarithmic and exponential equations
The function is given. You need to know how to use it to answer the question.
Example 1: The population of Russia can be described by the exponential function
P(t) = 142.9e—0.004t, where P(t) is the population, in millions, t years after 2006. When can
we expect the population of Russia to be 136 million?
Example 2: The formula A=Pert describes the accumulated value, A, of a sum of money, P,
the principal, after t years at an annual percentage rate of r (in decimal form) if the
interest is compounded continuously. How long will it take $5000 to double in value if the
money is invested at 7% compounded continuously?
Example 3: Students in a psychology class took a final exam. As part of an experiment to
see how much of the course content they remembered over time, they took equivalent
forms of the exam in monthly intervals thereafter. The average score, f(t), for the group
after t months is modeled by the function f(t) = 76 — 18log(t + 1), where 0 ≤ t ≤ 12.
When would we expect the group get an average score of 65?
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Example 4: The function P ( t ) = 107.4e0.012t models the population of Mexico, P(t), in millions, t
years after 2003. According to the model, when will Mexico’s population be 150 million?
Round to the nearest whole number.
Example 5: The function f(x) = 59.1 + 75.3lnx models the number of weight-loss
surgeries, f(x), in thousands, x years after 2001. According to this model, when will the
number of weight-loss surgeries reach 250 thousand?
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