1. Logarithmic Equations
Mrs.Volynskaya
Pre-Calculus Chapter 3.4
Exponential and
Logarithmic Equations
Properties of Exp. and Log Functions
loga ax = x
ln ex = x

2. Solving Exponential Equations
Take the ln of both sides.
Ex. ex = 72
ln ex = ln 72
x = ln 72
Ex. 4e2x = 5

3. Solving an Exponential Equation
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
= 3.417

4. Ex. e2x – 3ex + 2 = 0
(ex)2 – 3ex + 2 = 0
This factors.
( ) ( ) = 0
ex – ex - 1
Set both = 0 and finish
solving.
ex = 2
ex = 1
ln ex = ln 2
ln ex = ln 1
x = ln 2
x = 0

5. Ex. 2x = 10
ln 2x = ln 10
x ln 2 = ln 10
Ex. 4x+3 = 7x
ln 4x+3 = ln 7x
(x + 3) ln 4 = x ln 7
x ln ln 4 = x ln 7
Collect like terms
3 ln 4 = x ln 7 - x ln 4
Factor out an x
3 ln 4 = x( ln 7 – ln 4)

6. Solving a Logarithmic Equation
Ex. ln x = 2
Take both sides to the e
eln x = e2
x = e2
Ex ln x = 4
2 ln x = -1
ln x =
Ex. 2 ln 3x = 4
ln 3x = 2
3x = e2

7. Ex. ln (x – 2) + ln (2x – 3) = 2 ln x
+ means mult.
ln (x – 2)(2x – 3) = ln x2
e to both sides
to get rid of ln’s.
2x2 – 7x + 6 = x2
x2 – 7x + 6 = 0
( ) ( ) = 0
x – x – 1
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!