This document discusses exponential and logarithmic equations. It introduces e as a fixed number that exponential expressions approach as the number of compounding periods increases, defines natural logarithms with base e, and provides examples of solving various exponential and logarithmic equations by manipulating logarithmic expressions, isolating the variable, and applying inverse functions. Key steps include rewriting exponential expressions as logarithmic expressions, using logarithmic properties to combine or separate terms, and solving the resulting equation for the variable.

1. Exponential and Logarithmic
Equations
Chapter 4 Inverse, Exponential, and Logarithmic
Functions

2. Concepts and Objectives
⚫ Exponential and Logarithmic Equations
⚫ Identify e and ln x.
⚫ Set up and solve exponential and logarithmic
equations.

3. e
⚫ Suppose that $1 is invested at 100% interest per year,
compounded n times per year.
⚫ According to the formula, the compound amount at
the end of 1 year will be
⚫ As n increases, the value of A gets closer to some fixed
number, which is called e.
⚫ e is approximately 2.718281828.
⚫ Blind Date
1
1
n
A
n
 
= + 
 

4. Natural Logarithms
⚫ Logarithms with base e are called natural logarithms,
since they occur in the life sciences and economics in
natural situations that involve growth and decay.
⚫ The base e logarithm of x is written ln x.
⚫ Therefore, e and ln x are inverse functions.
ln
lnx x
e e x= =

5. Exponential Equations
⚫ Example: Solve . Round to the nearest
thousandth.
If x > 0, y > 0, a > 0, and a  1, then
x = y if and only if logax = logay
− +
=2 1 2
3 0.4x x

6. − = +2 log3 log3 log0.4 2log0.4x x
Exponential Equations
⚫ Example: Solve . Round to the nearest
thousandth.
If x > 0, y > 0, a > 0, and a  1, then
x = y if and only if logax = logay
− +
=2 1 2
3 0.4x x
− +
=2 1 2
log3 log0.4x x
( ) ( )=− +2 1 2log3 log0.4x x
− = +log0.l 4og32 log3 2log0.4x x
− = +log0.42 log3 2log g0.4 lo 3x x
( )− = +2log3 log0.4 2log0.4 log3x
+
=  −
−
2log0.4 log3
0.236
2log3 log0.4
x
− = +2 log3 log0.4 2log0.4 log3x x

7. Properties of Logs, Revisited
⚫ You could also finish this as
+
=
−
2log0.4 log3
2log3 log0.4
x
+
=
−
2
2
log0.4 log3
log3 log0.4
( )=
 
 
 
log 0.16 3
9
log
0.4
=
log0.48
log22.5
This is an
exact answer.

8. Solving a Logarithmic Equation
⚫ Example: Solve ( ) ( )log 6 log 2 logx x x+ − + =

9. Solving a Logarithmic Equation
⚫ Example: Solve ( ) ( )log 6 log 2 logx x x+ − + =
6
log log
2
x
x
x
+
=
+
6
2
x
x
x
+
=
+
( )6 2x x x+ = +
2
6 2x x x+ = +
2
6 0x x+ − =
( )( )3 2 0x x+ − =
3, 2x = −
= + −2
0 6x x
If we plug in –3, x+2
is negative. We can’t
take the log of a
negative number, so
our answer is 2.

10. Solving a Base e Equation
⚫ Example: Solve and round your answer to the
nearest thousandth.
2
200x
e =

11. Solving a Base e Equation
⚫ Example: Solve and round your answer to the
nearest thousandth.
2
200x
e =
2
ln ln200x
e =
2
ln200x =
ln200x = 
2.302x  
=
2
l 00ln n2x
e

12. Solving a Base e Equation
⚫ Example: Solve ( )ln
ln ln 3 ln2x
e x− − =

13. Solving a Base e Equation
⚫ Example: Solve ( )ln
ln ln 3 ln2x
e x− − =
( )− − =ln ln 3 ln2xx
( )− − =ln
ln ln 3 ln2x
xe
ln ln2
3
x
x
=
−
2
3
x
x
=
−
2 6x x= −
6 x=
( )= −2 3x x

14. Classwork
⚫ College Algebra
⚫ Page 464: 12-20 (even), page 442: 28, 30, 60-70
(even), page 429: 80-84 (even)