5.6 Exponential and logarithmic equations
How to solve exponential equations
(1) get equation into form:
a(expression with x in it) = (expression without x)
(2) take log or ln of both sides
(3) apply inverse property:
loga ax = x
or log of a power property: logb ax = x logb a
(4) solve for x
Example:
3x - 2
(1) e
= 5
(2) take logs:
(3) inverse prop.:
(4) solve:
e
3x - 2
-5=0
3x - 2
ln e
= ln 5
3x - 2
= ln 5
x = (ln 5 + 2)/3  1.20
Example: 10(2.5)3x - 17 = 0
(1) (2.5)3x = 17/10
(2) log (2.5)3x = log (17/10)
(3) 3x log 2.5 = log(17/10)
(log of a power property)
(4) x = log(17/10)/(3log 2.5)  .193
5.6-1
How to solve logarithmic equations
(1) isolate all the logs on left side, everything else on right
(2) write left-hand side as the log of a single expression
(collect) to get into form:
log (expression involving x) = expression with no x
(3) exponentiate both sides (raise base to powers of sides)
(4) use inverse property: a log a x  x
(5) solve for x
(6) check –– there may be extraneous roots!
Example:
ln (x + 3) + ln x - 1 = 0
(1) ln (x + 3) + ln x = 1
(2) ln (x + 3)x = 1
(3) eln(x + 3)x = e1
(4) (x + 3)x = e
(isolate)
(collect)
(exponentiate)
(inverse rule)
(5) solve quadratic equation x2 + 3x - e = 0
 3  9  4e
to get:
= .729, -3.729
2
solution with (-) choice of  is -, so is extraneous (why?)
(6) ANS: .729
5.6-2
Applications
Recall the formula for compound interest
 r
A = P 1 
 n
nt
Remember Hillary Hoghugger?
Her situation: She has $10,000 to invest in Arkansas
razorback pig futures that is projected to accumulate
earnings at 20% per year compounded quarterly.
Her question: How long will it take to double my
investment? Everything but t is known.
20000 = 10000(1.05)4t
find t
(1.05)4t = 2
log (1.05)4t = log 2
4t log (1.05) = log 2
log 2
t=
= .30103 = 3.55
.0848
4 log 1.05
Answer: 3.55 years
Mathematically:
Problem:
Solution:
It’s even easier if compounding is continuous:
A = Pert
20000 = 10000e.20t
e.20t = 2
.20t = ln 2
t = (ln 2)/.2 = 3.47 years
5.6-3