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MA 15800
Lesson 17
Exponential and Logarithmic Equations
Summer 2016
We have already discussed one method of how to solve an exponential equation; write each side
with the same base. (Or, perhaps you have discussed this method in previous math classes.)
Ex 1: Solve: 81x2  2753 x Both 81 and 27 can be written as powers of the number 3.
81x  2  2753 x
(34 ) x  2  (33 )53 x
34 x 8  3159 x If the bases are equal, the exponents are equal.
4 x  8  15  9 x
13x  23
x
Ex 2: Solve:
64
1
x 5
2
23
13
 163 x 1
Another method that can be used to solve exponential equations is to take a logarithm of the
same base of both sides. Examine the following.
Ex 3:
I will take the common logarithm of both sides.
3x  25
x
log 3  log 25
x log 3  log 25
log 25
x
 2.9299 to 4 decimal places
log 3
Notice that the same approximation is derived if the natural logarithm if used.
3x  25
ln 3x  ln 25
x ln 3  ln 25
x
ln 25
 2.9299
ln 3
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MA 15800
Lesson 17
Exponential and Logarithmic Equations
Summer 2016
Find exact answer and an approximation to 4 decimal places for the solutions to these equations.
Ex 4: 3 x  8
Ex 5: 3x4  213 x
Let’s return to example 3 and solve a different way.
3x  25 Convert to logarithmic form.
log3 25  x
We now have 3 different exact answers for x, the solution to the example
3 equation.
x  log3 25 
log 25 ln 25

log 3
ln 3
This statement leads to a theorem called the Change of Base Formula.
Theorem: Change of Base Formula
If 𝑢 > 0 and if a and b are positive real numbers different from 1, then
logb u 
log a u
log a b
In words: A logarithm equals the quotient of the logarithm of any positive real number base of
the argument and the logarithm (of the same base) of the original base.
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MA 15800
Lesson 17
Exponential and Logarithmic Equations
Summer 2016
Because our TI-30XA calculator has keys for natural logarithms (LN) and common logarithms
(LOG), here is the change of base formula using those bases.
logb u 
ln u
log u
and log b u 
ln b
log b
Ex 6: Use the change of base formula and your TI-30XA calculator to approximate the
following to four decimal places.
a) log 2 7 
b) log9 2.3 


Ex 7: Use the change of base formula and find the EXACT value for these logarithms.
log 5 16
log 4 32 
log 27 9 

log 5 64
The change of base formula could also be used to solve some basic exponential functions. Let’s
look back at example 3. 3x  25 In example 3 we solve this equation by first finding the
common logarithm (log) of both sides and second by finding the natural logarithm (ln) of both
sides. We could also solve this equation by using the change of base formula as seen here.
Ex 8:
3x  25
Convert to logarithmic form.
log 3 25  x Use the change of base formula on the left.
I used common logarithms.
log 25
x
log 3
x  2.9299
Notice this is the same as an answer found in example 3.
Ex 9: Solve by converting to logarithmic form and using the change of base formula. Give an
exact answer and an approximation to 4 decimal places.
(a)
(b)
12x2  15
4x  52
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MA 15800
Lesson 17
Exponential and Logarithmic Equations
Summer 2016
Ex 10:Use the change of base formula to approximate the y-intercept for the graph of the
following function. Round to 3 decimal places. Write your answer as an ordered pair.
f ( x)  log3 ( x  4)
Procedures for Solving Logarithmic Equations:
1.
If the equation can be written with a logarithm with the same base on both sides of the
equation, then the arguments are equal.
If logb M  logb N  M  N
2.
If the equation can be written with a logarithm on one side of the equation and a number
on the other, convert the equation to exponential form and solve.
If logb M  n where n is a real number, bn  M
Solve each equation.
log6 (2 x  3)  log6 12  log6 3
Ex 11:
Ex 12:
log x  log( x  1)  3log 4
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MA 15800
Lesson 17
Exponential and Logarithmic Equations
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2
Ex 13:
ln x  ln( x  6)  ln 9 (Check these solutions!)
Ex 14:
log3 ( x  3)  log3 ( x  5)  1
Ex 15:
ln x  1  ln( x  1)
Ex 16:
Summer 2016
log( x2  4)  log( x  2)  2  log( x  2)
(Round the solution to 2 decimal places.)
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MA 15800
Ex 17:
Ex 18:
Lesson 17
Exponential and Logarithmic Equations
Summer 2016
Solve A  Per t for r using common logarithms.
Solve the following equation without using a calculator.
log( x2 )  (log x)2
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