Solve an exponential equation using base 10 logarithms. Find the missing part of a logarithm equation. Use properties of logarithms to solve problems.

1. 4.3 Logarithmic Functions
Chapter 4 Inverse, Exponential, and Logarithmic
Functions

2. Concepts & Objectives
⚫ Logarithmic Functions
⚫ Solve an exponential equation with any positive base,
using base 10 logarithms
⚫ Use the definition of logarithm to find the logarithm,
base, or argument, if the other two are given.
⚫ Use the properties of logarithms to transform
expressions and solve equations.

3. Exponents Revisited
⚫ Consider the graph of the function f (x) = 10x.
What if I wanted to know
what x is when y is 40?
From the graph, it looks
to be 1.6, but plugging it
into the calculator, I find
101.6  39.811, not 40.

4. Exponents Revisited (cont.)
⚫ To solve 10x = 40, we can try to narrow it down by
plugging in different values:
⚫ Fortunately, our calculators have a function that does all
this for us: the logarithm function.
x 10x
1.61 40.74
x 10x
1.61 40.74
1.601 39.90
x 10x
1.61 40.74
1.601 39.90
1.602 39.99
x 10x
1.61 40.74
1.601 39.90
1.602 39.99
1.6021 40.00

5. Base 10 Logarithm
⚫ The inverse of an exponent is the logarithm (which is a
combination of “logical arithmetic”). The “base 10
logarithm” of a number is the exponent in the power of
10 which gives that number as its value.
y = log x if and only if 10y = x
log 10x = x
Inverse functions!

6. Base 10 Logarithm (cont.)
⚫ Example: Solve for x: 10x = 457

7. Base 10 Logarithm (cont.)
⚫ Example: Solve for x: 10x = 457
10x = 457
log 10x = log 457
x = 2.6599162…

8. Logarithms With Other Bases
⚫ Although we’ve been looking at powers of 10, the
concept of logarithms will work with any power. The
most important thing for you to remember to
understand logarithms is:
⚫ For example, can be rewritten as
A logarithm is an exponent.
2
log 32 5
= =
5
2 32

9. Logarithms
⚫ The formal definition would be:
⚫ To solve log problems, remember that the log is the
inverse of the exponent. To “undo” a log with a given
base, turn both sides of the equation into exponents of
that base.
⚫ You can also rewrite the equation into an exponent one.
y = logb x if and only if by = x
where x > 0, b > 0, and b  1

10. Examples
1. Find x if log3 x = –4.
2. Find x if log28 = x.

11. Examples
1. Find x if log3 x = –4.
2. Find x if log28 = x.
4
3 x
−
=
4
1 1
3 81
x = =
2 8
x
=
3
2 2
x
=
3
x =

12. Examples (cont.)
3. Find x if
2
log 4
3
x =

13. Examples (cont.)
3. Find x if
2
log 4
3
x =
2
3
4
x =
( )
3
2 3
2
3 2
4
x
 
=
 
 
8
x =

14. Properties of Logarithms
⚫ Because logarithms are exponents, they have three
properties that come directly from the corresponding
properties of exponentiation:
Exponents Logarithms
a b a b
x x x +
=
a
a b
b
x
x
x
−
=
( )
b
a ab
x x
=
( )
log log log
a b a b
= +
log log log
a
a b
b
 
= −
 
 
log log
b
a b a
=

15. Examples
1. Write log224 – log28 as a single logarithm of a single
argument.
2. Use the Log of a Power Property to solve 0.82x = 0.007.

16. Examples
1. Write log224 – log28 as a single logarithm of a single
argument.
2. Use the Log of a Power Property to solve 0.82x = 0.007.
2 2 2
24
log 24 log 8 log
8
 
− =  
 
2
log 3
=
2
log0.8 log0.007
x
=
2 log0.8 log0.007
x =
log0.007
11.12
2log0.8
x = 
=
2
0.8 0.007
x

17. Classwork
⚫ College Algebra
⚫ Page 442: 14-26 (even), page 429: 72-78 (even),
page 414: 78-86, 92, 94 (even)
⚫ 4.3 Classwork Check
⚫ Quiz 4.2