solve ex eq with logs

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Exponential and Logarithmic Equations
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Our first question then must be
What is a logarithm ?
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Definition of Logarithm
Suppose bgt0 and b?1, there is a number p such
that
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You must be able to convert an exponential
equation into logarithmic form and vice versa.
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Logarithmic Abbreviations

• log10 x log x (Common log)
• loge x ln x (Natural log)
• e 2.71828...

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Converting between Exponents Logarithms

• BASEEXPONENT POWER
• 42 16
• 4 is the base. 2 is the exponent.
  16 is the power.
• As a logarithm, logBASEPOWEREXPONENT
• log 4 16 2

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Example 1
Solution
We read this as the log base 2 of 8 is equal to
3.
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Example 1a
Solution
Read as the log base 4 of 16 is equal to 2.
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Example 1b
Solution
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Okay, so now its time for you to try some on
your own.
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Solution
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Solution
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Solution
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Solve Exponential Equations with Like Bases

• In an Exponential Equation, the variable is in
  the exponent. There may be one exponential term
  or more than one, like
• If you can isolate terms so that the equation can
  be written as two expressions with the same base,
  as in the equations above, then the solution is
  simple.

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Solve Exponential Equations with Like Bases

• Example 1 - One exponential expression.

1. Isolate the exponential expression and
rewrite the constant in terms of the same base.
2. Set the exponents equal to each other (drop
the bases) and solve the resulting equation.
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Solve Exponential Equations with Like Bases

• Example 2 - Two exponential expressions.

1. Isolate the exponential expressions on either
side of the . We then rewrite the 2nd
expression in terms of the same base as the first.
2. Set the exponents equal to each other (drop
the bases) and solve the resulting equation.
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Exponential Equations with Different Bases

• The Exponential Equations below contain
  exponential expressions whose bases cannot be
  rewritten as the same rational number.
• The solutions are irrational numbers, we will
  need to use a log function to evaluate them.

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Steps for solving exponential equations

• Take a common logarithm or Natural Log of each
  side
• Use the power property of logarithms
• Solve for x by dividing
• Use a calculator to find the approximate value

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Solving Exponential Equations
Solve . Round to the nearest
ten-thousandth.
1. Take the log of both sides
2. Use the power property
3. Solve for x.
X1.2619
4. Use a calculator.
Check your answer 31.26194
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Another Example
Solve . Round to the nearest
ten-thousandth.
1. Take the log of both sides
2. Use the power property
3. Solve for x.
X4.2009 4 0.2009
4. Use a calculator.
Check your answer 30.20094101
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Lets try some
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Lets try some
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Lets try some
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Lets try some
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Using Natural Logarithms to Solve Exponential
Equations
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Example Solving

• 2x 7 problem
• ln2x ln7 take ln both sides
• xln2 ln7 power rule
• x divide to solve for x
• x 2.807

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Solving exponential equations with a graphing
calculator

1. Type two equations into y

2. Graph. Suggest Zoom fit (0) especially for
large values
3. Use the calc function to find the intersection
of the two graphs.
Solution 2.0408