Logarithms Tutorial
Understanding the Log Function
March 2003
S. H. Lapinski
Where Did Logs Come From?
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The invention of logs in the early 1600s fueled the
scientific revolution. Back then scientists,
astronomers especially, used to spend huge amounts
of time crunching numbers on paper.
By cutting the time they spent doing arithmetic,
logarithms effectively gave them a longer productive
life.
The slide rule was nothing more than a device built
for doing various computations quickly, using
logarithms.
There are still good reasons for
studying them.
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To model many natural processes, particularly
in living systems. We perceive loudness of
sound as the logarithm of the actual sound
intensity, and dB (decibels) are a logarithmic
scale.
To measure the pH or acidity of a chemical
solution.
To measure earthquake intensity on the
Richter scale.
How they are developed
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In the mathematical operation of addition we take
two numbers and join them to create a third 4 + 4 =
8
We can repeat this operation: 4 + 4 + 4 = 12
Multiplication is the mathematical operation that
extends this: 3 • 4 = 12
In the same way, we can repeat multiplication: 3 • 3
• 3 = 27
The extension of multiplication is exponentiation: 3 •
3 • 3 = 27 = 33
More on development
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The exponential function y = 2x is shown in this
graph:
More on development
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Now consider that we have a number and we want to know
how many 2's must be multiplied together to get that number.
For example, given that we are using `2' as the base, how
many 2's must be multiplied together to get 32? That is, we
want to solve this equation: 2B = 32
Of course, 25 = 32, so B = 5. To be able to get a hold of this,
mathematicians made up a new function called the logarithm:
log2 32 = 5
Inverses
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This graph was created by switching the x and y of the
exponential graph, which is the same as flipping the curve
over on a 45 degree line.
2log2 a  a
and log 2 2a  a
DEFINITION:
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The base a logarithm function y = logax is
the inverse of the base a exponential
function y = ax (a > 0, a  1)
How to Convert Between
Different Bases
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Calculators and computers generally don't calculate the
logarithm to the base 2, but we can use a method to make
this easy.
Take for example, the equation 2x = 32. We use the change
of base formula!! We can change any base to a different
base any time we want. The most used bases are obviously
base 10 and base e because they are the only bases that
appear on your calculator!
Loga x
Logb x 
Loga b
Pick a new base and the formula says it is equal
to the log of the number in the new base
divided by the log of the old base in the new base.
Examples
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Find the value of log2 37
Change to base 10 and use your calculator. log 37/log 2
Now use your calculator and round to hundredths.
= 5.21
Log7 99 = ?
Change to base 10 or base e. Try it both ways and see.
log3 81
log4 256
log2 1024
Properties Of Logarithms
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For any real numbers x > 0 and y >0,
 Product Rule: loga xy = loga x + loga y
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x
 Quotient Rule: log a  log a x  log a y
y
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 Power Rule: loga xy = yloga x
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More Practice
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Express each as a single log.
Log x + Log y - Log z =
2 Ln x + 3 Ln y =
Solve
Log2 (x + 1) + Log2 3 = 4
Log (x + 3) + Log x = 1
Web Sites
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http://www.shodor.org/UNChem/math/logs/
http://www.physics.uoguelph.ca/tutorials/LOG/
http://www.purplemath.com/modules/logs.htm
http://www.exploremath.com/activities/Activity_p
age.cfm?ActivityID=7
SAMPLE Test on Logs
http://www.alltel.net/~okrebs/page58.html