Logarithms
ISP 121
What is a Logarithm?
• A logarithm (or log) is a number that
represents a power or exponent
• Why use logs?
– A simpler way to express large values
– Some things grow or shrink exponentially, so
the log is a perfect “numbering” system
What is a Logarithm?
• Logarithms are exponents in disguise.
• The output of a logarithm is an exponent
log b X = Y
↔
bY = X
• The first equation is in “log form” and the
second equation is in “exponent form”
Converting and Solving
log3 27 = y
3y = 27
y=3
So log3 27 = 3
log525 = y
5y = 25
Y=2
log525 = 2
log7 49 = y
7y = 49
y=2
So log7 49 = 2
log5125 = y
5y = 125
Y=3
log5125 = 3
Log Base 10
• The log base 10 (written log10 or
just log) is a very common log
• log10 x is the power to which 10
must be raised to obtain x
• Or better yet, 10 to what power
equals x?
Log Base 10
Solve the following without a calculator:
log10 100 = y
10y = 100
102 = 100
log10 100 = 2
Log10 1000 = y
10y = 1000
103 = 1000
log10 1000 = 3
log10 10,000 = y
104 = 10,000
log10 10,000 = 4
Log10 10,000,000 = y
107 = 10,000,000
log10 10,000,000 = 7
Log Base 10
Solve the following with a calculator:
log10 3 = y
log10 3 = .4771
Log10 30 = y
log10 30 = 1.4771
Log10 300 = y
log10 300 = 2.4771
Review the answers.
With log base 10, how
do the answers
change when x is
multiplied by 10?
y increases by 1
For log base 10, when
y increases by 1, x is
multiplied by 10
Log Base 10
Use what we learned in the last slide.
Remember, log = log10
log 2.5 = y
Log 2.5 = .39794
So what is log 25?
Log 250?
Log 25000?
log 25 = 1.39794
log 250 = 2.39794
log 2500 = 4.39794
Log x = 1
101 = 10
X = 10
So what is x if log x = 2?
what is x if log x = 3?
what is x if log x = 4?
what is x if log x = 6?
log x = 2 x = 100
log x = 3 x = 1000
log x = 6 x = 100,000