Logarithms – An Introduction
Check for Understanding –
3103.3.16
Check for Understanding
– 3103.3.17
Prove basic properties of
logarithms using properties
of exponents and apply
those properties to solve
problems.
Know that the logarithm and
exponential functions are
inverses and use this information
to solve real-world problems.
What are logarithms?
log·a·rithm : noun
the exponent that indicates
the power to which a base
number is raised to produce
a given number
Merriam-Webster Online (June 2, 2009)
What are logarithms used for?
• pH
Scale
• Telecommunication
• Richter Scale
• Electronics
• Decibels
• Optics
• Radioactive Decay
• Astronomy
• Population Growth
• Computer Science
• Interest Rates
• Acoustics
… And Many More!
My calculator has a log button…
why can’t I just use that?
The button on your
calculator only works
for certain types of
logarithms; these are
called common
logarithms.
Try These On Your Calculator
log245
1.6532
X
5.4919
P
log10100
2
P
What’s the difference?
The log button on the calculator is used
to evaluate common logarithms, which
have a base of 10.
If a base is not written on a logarithm,
the base is understood to be 10.
log 100 is the same as log10100
The logarithmic function is an inverse of
the exponential function.
Logarithm with base b
The basic mathematical definition of
logarithms with base b is…
y
logb x = y iff b = x
b > 0, b ≠ 1, x > 0
Write each equation in exponential form.
1. log6 36 = 2
2
6 = 36
2. log125 5 = 1
5
1
5
125 = 5
Write each equation in logarithmic form.
3. 23 = 8
log2 8 = 3
-2
4. 7 = 1
49
1
log
7 49
= –2
Evaluate each expression
5. log4 64 = x
x
4 = 64
x
6. log5 625 = x
x
5 = 625
x
4
4 =4
5 =5
x=3
x=4
3
Evaluate each expression
7. log2 128
1
8. log3
81
9. log8 4
10. log11 1
Evaluate each expression
7. log2 128
7
1
8. log3
81
–4
9. log8 4
⅔
10. log11 1
0
Solve each equation
11. log4 x = 3
3
4 =x
12. log4 x = 3
2
3
2
4 =x
64 = x
x=8
Evaluate each expression
13. log6 (2y + 8) = 2
15. log7 (5x + 7) = log7 (3x + 11)
14. logb 16 = 4
16. log3 (2x – 8) = log3 (6x + 24)
Evaluate each expression
13. log6 (2y + 8) = 2
14
14. logb 16 = 4
2
15. log7 (5x + 7) = log7 (3x + 11)
2
16. log3 (2x – 8) = log3 (6x + 24)
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