Algebra II/Trig Honors
Unit 4 Day 7 Notes: Solve Exponential and Logarithmic Equations
Objective: To solve exponential and logarithmic equations
Definition:

Exponential Equation - __________________________________________________________
Property for Equality of Exponential Equations:
o If b is a positive number other than 1, then b x  b y if and only if x  y
What that looks like: If 3 x  35 , then x  _______
o Goal when solving: _______________________________________________________
Example 1: Solve by equating exponents
Solve.
a.
1
4  
2
x 3
x
b. 9 2 x  27 x 1
Practice: Solve.
a. 100 7 x 1  1000 3 x  2

1
b. 813 x   
3
5 x 6
If you cannot get the bases to match, ________________________________________________
______________________________________________________________________________
Example 2: Take a logarithm of each side
Solve.
a. 4 x  11
b. 7 9 x  15
Practice: Solve.
b. 4e 0.3 x  7  13
a. 2 x  5
Application of Exponential Equations: Newton’s Law of Cooling
T  T0  TR e  rt  TR

T0 = initial temperature




T = temperature after a certain period of time
t = time (in minutes)
TR = surrounding temperature
r = substance’s cooling rate
Example 3: Use an exponential model
You are driving on a hot day when your car overheats and stops running. It overheats at 280F and can
be driven again at 230F . If r  0.0048 and it is 80F outside, how long (in minutes) do you have to
wait until you can continue driving?
Definition:

Logarithmic Equation - __________________________________________________________
Property for Equality of Logarithmic Equations:
o If b, x, and y are positive numbers with b  1, then log b x  log b y if and only if x  y
What that looks like: If log 2 x  log 2 7 , then x  _______
o Goal when solving: _______________________________________________________
Example 4: Solve a logarithmic equation
Solve log 5 4 x  7   log 5 x  5
Practice: Solve ln 7 x  4  ln 2 x  11
Example 5: Solve a logarithmic equation using exponential form
Solve log 4 5x  1  3
Practice: Solve log 2 x  6  5
*Since logarithmic functions generally have a limited domain, we must check for __________________
________________ by checking your answer(s) in the original equation.
Example 6: Solve logarithmic equations using logarithmic properties
Solve log 2x  log x  5  2
Practice: Solve log 4 x  12  log 4 x  3
Example 7: Use a logarithmic model
The apparent magnitude of a star is a measure of the brightness of the star as it appears to observers on
Earth. The apparent magnitude M of the dimmest star that can be seen with a telescope is given by the
function M  5 log D  2 where D is the diameter (in millimeters) of the telescope’s objective lens. If a
telescope can reveal stars with a magnitude of 12, what is the diameter of its objective lens?
HW: Page 271 #3-42 (M3), 44, 48, 50, 52, 57