2-3 Part II: Solving Exponential and Logarithmic
Equations
Recall: log and exponent functions are inverses; that is:
if you
if you
 start with the number 5
 raise e to the 5th power
 then take the loge (ln) of the
result
 you just end up with the
original 5
 try it on your calculator!




start with the number 5
take the loge (ln) of it
then raise e to that power
you just end up with the
original 5
Mathematically, ln e5 = 5 and eln 5 = 5.
In general, we have the inverse properties of exponentiation
and logs:
ln ex = x
and
eln x = x
These properties are indispensable for the methods to be shown.
2-3 Part II
p. 1
How to solve exponential equations
 how do you solve equations like:
e
= 5 ?
 methods we know won't work, because x is in an exponent!
 how can we get x out of the exponent, so we can solve as a
non-exponential equation?
3x - 2
The idea: Isolate exponential and use logs
Example:
Solve for x:
e
3x - 2
1. take ln of each side
(base is e)
2. apply inverse property
= 5
ln e
3x - 2
= ln 5
3x - 2 = ln 5
note: steps 1 and 2 can be shortcut into one step
3. solve
Example:
x = (ln 5 + 2)/3 = 1.203
Solve I = 1 - e-Rt/L
1. isolate exponential
(for t)
e-Rt/L = 1 - I
2. take logs/use inverse property -Rt/L = ln (1 - I)
3. solve
2-3 Part II
t = -L ln(1 - I)/R
p. 2
How to solve logarithmic equations
 how do you solve equations like: ln x = (2/3)ln 8 + (1/2)ln 9 ?
 methods we know won't work, because x is in an argument of
a log!
 how can we get x out of the ln, so we can solve as a nonlogarithmic equation?
The idea: Use contraction, then one-to-one property or
exponentiation!
Example: ln x = (2/3)ln 8 + (1/2)ln 9
contract right-hand side
ln x = ln 12
use one-to-one (eliminate ln)
Example:
contract
x = 12
log (x + 3) - log x = 1
log((x+3)/x) = 1
exponentiate (raise 10 to lhs and rhs of
equation)
(x+3)/x = 10
solve
x = 1/3
2-3 Part II
p. 3