Download Intermediate Algebra Section 9.5 – Exponential and Logarithmic

Intermediate Algebra
Section 9.5 – Exponential and Logarithmic Equations
The following properties result from the fact that exponential and
logarithmic functions are one-to-one. These properties will be used to
solve equations that involve exponential or logarithmic functions.
One-to-One Property of Logarithms
In the following property, M , N , and a are positive real numbers,
with a ≠ 1 .
If log a M = log a N , then M = N .
Example:
Solve the following equations. Express irrational
solutions in exact form and as a decimal rounded to three decimal
places.
a)
log 5 x = log 5 13
b)
1
ln x = 2 ln x
2
To solve a logarithmic equation when the One-to-One Property of
Logarithms can’t be used, first isolate the logarithmic expression,
then exponentiate each side of the equation (or write the equation in
exponential form) and solve for the variable. We must always check
our solutions in the original equation to ensure that we do get an
extraneous solution.
Section 9.5– Exponential and Logarithmic Equations
Example:
Solve the following logarithmic equations.
a)
log 6 (x 2 − x )= 1
b)
log 4 10 − log 4 x = 2
c)
log 3 x + log 3 (x + 6) = 3
page 2
Section 9.5– Exponential and Logarithmic Equations
d)
page 3
log 2 x − log 2 (3x + 5) = 4
Recall that we previously used the One-to-One Property of
Exponents to solve exponential equations.
Example:
Solve the following exponential equations.
3x
x −1
x
a) 2 = 16
b) 3 = 81
To solve an exponential equation when the One-to-One Property of
Exponents can’t be used, first isolate the exponential expression and
then take the logarithm of each side of the equation (or write the
equation in logarithmic form) and solve for the variable.
Example:
a)
2 = 39
x
Solve the following exponential equations.
b)
6
x +3
=2
Section 9.5– Exponential and Logarithmic Equations
c)
−4e x = −16
page 4
d)
10 y = 3.7
Example:
Based on data obtained from the Kelley Blue Book, the
value V of a Dodge Stratus that is t years old can be modeled
by V ( t ) = 19, 282 ( 0.84 ) .
a) According to the model, when will the car be worth $10,000?
b) According to the model, when will the car be worth $5,000?
c) According to the model, when will the car be worth $1,000?
t
Section 9.5– Exponential and Logarithmic Equations
page 5
Example:
A baker removes a cake from a 350 F oven and places
it in a room whose temperature is 72 F. According to Newton’s Law
of Cooling, the temperature u ( in F) of the cake at time t (in
minutes) can be modeled by u ( t ) = 72 + 278e −0.0835t . According to this
model, what will the temperature of the cake be
a) after 15 minutes?
b) after 30 minutes?