1. Exponentials Rules
Happy Tuesday 10-7

2. Exponentials For these rules to work they must have the same base.
𝑦=𝑎∗ 𝑏 𝑥
A is initial value.
B is growth factor, also known as the base

3. Solving for Exponent
When asked to solve an exponential equation such as 2 𝑥 + 6 = 32 the first thing we need to do  is to decide which way is the “best” way to solve the problem.

4. Solving for Exponent Some exponential equations can be solved by
rewriting each side of the equation using the same base others will need to use a logarithm.

5. Solving for Exponent When solving for exponents it is known
𝑖𝑓 𝐵 𝑚 = 𝐵 𝑛 𝑡ℎ𝑒𝑛 𝑚=𝑛
Therefore if we can manipulate one or both sides to have the same base with can easily solve for the missing exponent.

6. Break it down to its base
In order to break down or bring it up to a “base number” we need to know how many times the smallest base number in the original base repeats.
Example of bases are the prime numbers: 2,3,5,7,11.
6 can be included even though it isn’t prime and can 10.

7. Break it down to its base
Example:
15 𝑥 can’t be re-written as another whole number base 15=5∗3 nothing repeats
9 𝑥 can be re-written as another base, 9=3∗3, three repeats so 9 𝑥 ⇒ 𝑥 = 3 2𝑥

8. Good to know
It would be good to know the squares, cubes and quartic numbers of the first 10 digits
a) 1= 1 2 = 1 3 = 1 4
b) 2 1 , 2 2 =4, 2 3 =8, 2 4 =16
*hint 2 will be used a lot
Fill in the rest of the table:

9. Good to know 1st Squared Cubed Quartic 3 9 4 16 5 25 6 36 7 49 8 64 8110
100

10. Re-write the bases Re-write 128 as a base of 2. Solution: 2 7
2) Re-write: 9, 27 and as a base of 3
Solutions: 3 2 , , 3 −4
3) Re-Write: , 25, 625 as a base of 5
Solutions: 5 −3 , 5 2 , 5 4

11. Example
Solve for the missing Exponent: 2 𝑥 + 6 = 32 1st :re-write in terms of base 2 2 𝑥+6 = 2 5 2nd: since bases are now the same drop the base and solve for exponent. 𝑥+6=5 3rd: solve 𝑥=−1

12. You try Solve for x: 9 2𝑥−5 =27 Steps: 3 2 2𝑥−5 = 3 3 3 4𝑥−10 = 3 3
𝑥−5 = 3 3
3 4𝑥−10 = 3 3
4𝑥−10=3
Solution: 𝑥= 13 4

13. Practice solving
Practice solving by changing the base, please state if it isn’t possible to make the bases match.
12 𝑥−9 =144
𝑥+4 =1296
25 (−𝑥−1) =125
27 3𝑥+5 = 1 27

14. Quote of the week
“The journey of a thousand miles begin with one step.” - Lao Tzu