This document discusses common logarithms and how to evaluate logarithmic expressions with and without a calculator. It provides examples of rewriting exponential expressions as logarithmic expressions by setting them equal to variables and manipulating the equations. It also introduces the change of base formula for evaluating logarithms with bases other than 10.

1. Day 3
Common
Logarithms

2. Express each number using exponents.
O A. 36 G. 1
O B. 121 What about . . .
O C. 4 H. 345
O D. I. 0.0023
O E. 100
O F. 1000

3. Logarithms give you a way to
solve for an exponent.
O Ex: 5x = 12
Common Logarithms are any
logarithm of base 10
Ex: log10 or log

4. Rewriting:
10b = a log10a=b
A. 102 = 100
B. 33 = 27
C. 25 = 32
log10 100 = 2
log3 27 = 3
log2 32 = 5

5. E. 641/2 = 8
F. log6 36 = 2
Log64 8 = 1/2
D. 9-2 =
1
81
G. log3 81 = 4
H. log14 = -2
1
196
I. log10 10 = 1
J. Log 1 = 0
Log9 = -2
1
81
62 = 36
34 = 81
101 = 10
100 = 1
14-2 =
1
196

6. Evaluate without a calculator:
A. Log4 64
Step 1: set = to x Log4 64 = x
Step 2: rewrite in
exp. form
4x = 64
Step 3: break down
the #’s
22x = 26
Step 4: once the bases
are the same,
set exp. =
2x = 6
x = 3

7. B) Log5 125
5x = 125
5x = 55
x = 5
C) Log4 16
4x = 16
22x = 24
2x = 4
x = 2

8. D) Log343 7
343 = 7x
73x = 71
x = 1/3
Log343 7 = x
3x = 1

9. E) Log3
3x = 3-5
Log3 = x
x = -5
243
1
3x = 243
1
243
1

10. Evaluate using a calculator.
A) Log 75 1.8751
B) Log -3 Not possible
** log10 (-3) = x
10x = -3
C) Log 1 0
** log10 1= x 10x = 1

11. But what if the base isn’t 10?
Use Change of Base Formula:
logb a = or

12. A) log5 3 log 3
log 5
= 0.6826
B) log11 18 log 18
log 11
= 1.2054
C) log4 8
log 8
log 4 = 1.5