2. 5.7 – Exponential Equations; Changing Bases
Essential Question: How do you solve for a variable exponent when you cannot get the same base on both sides of the equal sign?

3. Exponential Equation 2x – 3 = 8 2x – 3 = 23
an equation that contains a variable in the exponent.
2x – 3 = 8
(Some cannot be solved this way.)
2x – 3 = 23

4. Change of Base Formula
log b c =
Ex. Find log 6 7.

5. Examples
a) 3-x = .7 b)
c) 3x = 9
d) 1.1x = 2

6. Example
The half-life of carbon-11 is 20 minutes. How long will it take for 800 g of carbon-11 to decay to:
a) 640 g
b) 40 g

7. Example
The population of Kenya reached 25,000,000 people in When will it reach 50,000,000 people? Assume an annual rate of increases of 4.1%.

8. Example
A $10,000 certificate of deposit at a certain bank will double in value in 9 years.
a) Give a formula for the accumulated amount t years after the investment is made.
b) How long does it take the money to triple in value?

9. Example
For each pair of equations, solve one of them by using powers of the same base number. To the nearest hundredth, solve the other by using logarithms.
a) 4x = 16
b) 25x =
4x = 20
25x = 2