Solving Exponential and Logarithmic Equations

a^x = a^y 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 5^x = 0.04 for x.

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

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

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

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.