Transcript Solve Exponential Equations Using Common Logs

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.
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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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?
We want to solve this equation: 2B = 32
25 = 32, so B = 5.
mathematicians made up a new function called the logarithm:
log2 32 = 5
DEFINITION:
a logarithm function
y = logax
is the inverse of
a exponential function
y = abx
If you want to undo a exponent, use a logarithm
Using Common Log to solve
exponential equations
1. 3x  27
2. 4  16
What power to I raise 3 to, to get 27?
x3
x
x2
3. 2x  32
x5
4. 6 x  1, 296
What power to I raise 4 to, to get 16?
What power to I raise 2 to, to get 32?
What power to I raise 6 to, to get 1,296?
x?
As you can see, these get difficult….
We can use common logs to solve them
Using Common Log to solve
exponential equations
we will start with one we know, write down the steps
1. 3  27
x log 3 = log 27
log 3 log 3
log 27
x =
log 3
x
x =3
1. Isolate the base with the exponent
2. Rewrite it in log notation
a. move the exponent out front
b. Write log in front of the two bases
3. Solve for x, divide both sides by
by everything but the x
4. Find the log button on the calculator
(it is next to the 7)
5. log(27) / log(3) enter
Using Common Log to solve
exponential equations
Now, a harder one…..
4. 6  1296
x log 6 = log 1296
log 6 log 6
log 1296
x =
log 6
x
x =4
1. Isolate the base with the exponent
2. Rewrite it in log notation
a. move the exponent out front
b. Write log in front of the two bases
3. Solve for x, divide both sides by
log 6
4. Find the log button on the calculator
(it is next to the 7)
5. log(1296) / log(6) enter
Using Common Log to solve
exponential equations
Now, a harder one…..
5. 34 x  3  6564
34 x  6561
4x log 3 = log 6561
4log 3 4log 3
log 6561
x =
4 log 3
x =2
1. Isolate the base with the exponent
2. Rewrite it in log notation
a. move the exponent out front
b. Write log in front of the two bases
3. Solve for x, divide both sides by
4log 3
4. log(6561) /(4 log(3) enter