This document discusses different methods for solving exponential and logarithmic equations: 1) If two powers have the same base, set their exponents equal to solve for the variable. Alternatively, take the log of both sides to rewrite in exponential form. 2) For natural log equations, take the ln of both sides to isolate the exponential term before solving for the variable. 3) Check that any solutions satisfy the original equation, as extraneous solutions can arise from the limited domains of log functions.

2. Solving Exponential Equations
 If two powers with the same base are
equal, then their exponents must be equal.
 To solve using this property:
 Rewrite equation so the bases match.
 Set exponents equal to each other.
 Solve for x.

3. Examples:
 Solve:

4. Your Turn!
 Solve

5. Solving by Taking Logs
 If it is not easy/possible to rewrite using
same bases, take the log (with base b) of
both sides.
 Examples:

6. Your Turn!
 Solve

7. More Examples: Taking Logs
 Not all equations will be that basic!
 Must get the exponential part by itself, first.
 Then, take the log of both sides.
 Last, continue solving for x.
 Example:

8. Example:
 Solve

9. Your Turn!
 Solve

10. Natural Base Exp. Equations
 Get the exponential part alone.
 Take the ln of each side.
 Example:

11. Exponential Models
 Newton’s Law of Cooling –
 the temperature T of a cooling substance is
modeled by
where:
t = time (in minutes)
T0 = initial temp. of substance
TR = room temp.
r = cooling rate

12. Example:
 Your mom is making soup for dinner. When
she takes it off the stove, it is 212ºF. The
room temp. is 70ºF and the cooling rate of the
soup is r = 0.046. How long will you have to
wait to eat the soup at a safe temp. of 100ºF?

13. Solving Log Equations
 For two logs with the same base:
 Example:
 Solve

14. Your Turn!
 Solve

15. Equations with One Log
 “Chop off the log” and write in exponential
form. Then solve for x.
 Example:
 Solve

16. Your Turn!
 Solve

17. Extraneous Solutions
 The domains of log functions are limited.
 This sometimes produces extraneous solutions.
 Always check answers in original equation.
 Example:
 Solve

18. Example:
 Solve

19. Your Turn!
 Solve