3.4
Exponential and
Logarithmic Equations
Properties of Exp. and Log Functions
loga
ln
ex
ax
=x
=x
a
e
log a x
ln x
=x
=x
Solving Exponential Equations
Ex. ex = 72
ln ex = ln 72
Take the ln of both sides.
x = ln 72 ! 4.277
Ex.
4e2x = 5
e
2x
ln e
5
=
4
2x
5
= ln
4
5
2 x = ln
4
1 5
x = ln ! .112
2 4
Solving an Exponential Equation
Ex.
2(32t-5) - 4 = 11
First, add 4 to each side
2(32t-5) = 15
Divide by 2
(32t-5) = 15/2
ln(32t-5) = ln 7.5
(2t-5) ln3 = ln 7.5
2tln3 - 5ln3 = ln 7.5
2tln3 = 5ln3 + ln 7.5
5ln 3 + ln 7.5)
(
t=
(2ln 3)
= 3.417
Ex.
e2x – 3ex + 2 = 0
(ex)2 – 3ex + 2 = 0
This factors.
( ex – 2 ) ( ex - 1 ) = 0
ex = 2
ln ex = ln 2
x = ln 2
ex = 1
ln ex = ln 1
x=0
Set both = 0 and finish
solving.
Ex.
2x = 10
ln 2x = ln 10
ln 10
x=
! 3.322
ln 2
x ln 2 = ln 10
Ex.
4x+3 = 7x
ln 4x+3 = ln 7x
3ln 4
x=
ln 7 " ln 4
! 7.432
(x + 3) ln 4 = x ln 7
x ln 4 + 3 ln 4 = x ln 7
!
3 ln 4 = x ln 7 - x ln 4
3 ln 4 = x( ln 7 – ln 4)
Collect like terms
Factor out an x
Solving a Logarithmic Equation
Ex.
ln x = 2
eln x = e2
x = e2
Ex.
Ex.
Take both sides to the e
! 7.389
5 + 2 ln x = 4
2 ln x = -1
1
ln x = !
2
2 ln 3x = 4
ln 3x = 2
3x = e2
x=e
!1
2
2
! .607
e
! 2.463
x=
3
Ex.
ln (x – 2) + ln (2x – 3) = 2 ln x
ln (x – 2)(2x – 3) = ln x2
2x2 – 7x + 6 = x2
+ means mult.
e to both sides
to get rid of ln’s.
x2 – 7x + 6 = 0
( x – 6 ) ( x – 1) = 0
6 and 1 are possible answers
Remember, can not take the log of a neg. number
or zero. Put answers back into the original to
check them.
Notice that only 6 works!