Solving Exponential and Logarithmic Equations

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Solving Exponential and Logarithmic Equations
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Exponential Equations are equations of the form
y abx. When solving, we might be looking for
the x-value, the b value or the y- value. First,
well review algebraic methods.
When solving for b, isolate the b value then
raise both sides of the equation to the
reciprocal power of the exponent.
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When solving for y, solve by performing the
indicated operations.
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When solving for the exponent, rewrite the bases
so they have the same base. If the bases are
equal, the exponents are equal. Now, solve.
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Solve the following exponential equations
3 b
y 25
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Solving Exponential Equations by Using the
Graphing Calculator

• Always isolate the variable FIRST!!
• Graph the function in Y1
• Graph the rest of the equation in Y2
• Use the intersect function (found with
  ) to determine the
• x value

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Solve for x to the nearest thousandth ex72

• Graph Y1 ex and Y2 72
• Use intersect to give the answer

x 4.277
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Log Equations are of the form ylogba or y ln x
where the base is e.
Y log216 4log3x 3 logx1000
To solve for x, we need to undo the log format by
rewriting in exponential form.
2y16 34x x31000
Now we use the exponential rules to solve.
2y16 81 x X3
103 y3
x4
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Solve algebraically
Rewrite as an ln equation
Since ln ex means the exponent of ln ex, just use
x
Note when an equation is written in terms of e,
you MUST use natural logs. Otherwise you may use
log or ln at will.
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Solve for x algebraically 2x 14
Take the ln or log of each side and solve.
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Solve for x 2x14 graphically.

• Graph Y1 2 x and Y2 14
• Use intersect to approximate the answer

X3.807
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Solve lnx 3 to the nearest tenth.
X 20.1
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Solve for x
X 5
X-2
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More Involved Equations
Sometimes log or ln equations require a few more
steps to clean them up before we can simply
take the log or undo the log.
Before you can take the ln, you need to isolate
it!
2ln x -1 ln x -0.5
e-0.5x x 0.60653
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To solve graphically, you still must isolate the
ln expression. 2ln x -1
Graph as y1 2lnx y2 -1
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More practice
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Solving Logarithmic Equations Algebraically Using
Laws of Logarithms
When an equation contains the word log or ln, we
need to eliminate it to solve the equation so
first we apply the laws of logarithms to undo
the addition by changing to multiplication,
undo subtraction by changing it to division,
and undo powers by changing them to
multiplication..
Solve Log 2(4x10) log2(x1) 3
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Log 2(4x10) log2(x1) 3
Apply Quotient Rule.
Definition of Logarithm
Cross multiply and solve
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• Solve

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