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Solving Exponential and Logarithmic Equations

Objective: In this lesson you learned how to solve exponential and
logarithmic equations.

I. Introduction What you should learn
How to solve simple
exponential and
logarithmic equations
State the One-to-One Property for exponential equations.

ax = ay if and only if x = y

State the One-to-One Property for logarithmic equations.

if and only if x = y

State the Inverse Properties for exponential equations and for
logarithmic equations.

Describe how the One-to-One Properties and the Inverse
Properties can be used to solve exponential and logarithmic
equations.

· Rewrite the given equation in a form that allows the use
of the One-to-One Properties of exponential or
logarithmic functions to solve the equation.
· Rewrite an exponential equation in logarithmic form and
apply the Inverse Property of logarithmic functions to
solve the equation.
· Rewrite a logarithmic equation in exponential form and
apply the Inverse Property of exponential functions to
solve the equation.

Example 1:

(a) Solve   for x.

(b) Solve 5x = 0.04 for x.

(a) x = 2 (b) x = - 2

 
II. Solving Exponential Equations What you should learn
How to solve more
complicated exponential
equations
Describe how to solve the exponential equation 10x = 90
algebraically.

Take the common logarithm of each side of the equation and
then use the Inverse Property to obtain: x = log 90. Then use a
calculator to approximate the solution by evaluating log 90 ≈
1.954.

Example 2: Solve e x-2 - 7 = 59 for x. Round to three decimal
places.
x ≈ 6.190

 
III. Solving Logarithmic Equations What you should learn
How to solve more
complicated logarithmic
equations
Describe how to solve the logarithmic equation
algebraically.

Use the One-to-One Property for logarithms to write the
arguments of each logarithm as equal: (4x - 7) = (8 - x). Then
solve this resulting linear equation by adding 7 to each side,
adding x to each side, and then finally dividing both sides by 5.
The solution is x = 3.

Example 3:
Solve 4 ln 5x = 28 for x. Round to three decimal
places.
x ≈ 219.327

 
IV. Approximating Solutions What you should learn
How to approximate the
solutions of exponential
or logarithmic equations
with a graphing utility
Describe at least two different methods that can be used to
approximate the solutions of an exponential or logarithmic
equation using a graphing utility.

Graph the left-hand side and the right-hand side of the equation
in the same viewing window; then use the intersect feature or the
zoom and trace features of the graphing utility to find the points
of intersection. OR rewrite the equation so that all terms on the
left side are equal to 0. Then use a graphing utility to graph the
left side of the equation. Use the zero or root feature or the zoom
and trace features to approximate the solutions of the equation.

 
V. Applications of Solving Exponential and Logarithmic
Equations
What you should learn
How to use exponential
and logarithmic equations
to model and solve reallife
problems
Example 4:
Use the formula for continuous compounding,
, to find how long it will take $1500 to
triple in value if it is invested at 12% interest,
compounded continuously.
t ≈ 9.155 years
 
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