change of base formula

1. Honors Math 3
Please turn in your test corrections
Warm up
Evaluate each expression:
49log7
001.log 23
3
1
loga. b. c.
d.
Solve
2 𝑥
= 4 𝑥−6

2. The Lame Joke of the day..
And now it’s time for..
How do we know that the following fractions
are in Europe? A/C, X/C, and W/C?
Because their numerators are all over C’s!

3. The Lame Joke of the day..
And now it’s time for..
Where does a Bee go to the bathroom?
A BP Station!

4. Learning Objectives
1. Use the change of base formula to convert
logarithm bases.
2. Use the change of base formula to solve an
exponential equation.
You will need a calculator today.
Cell phones are not allowed on a test, so please invest in a
calculator if you do not already have one.

5. Introduction
We can solve this easy without
a calculator:
x27log3
3x
What about this?
4log3
43 x
3 raised up to WHAT power gives 4?:

6. I dunno!

7. Can I solve it with
a calculator?
Yes you can!
With The Change
of Base
Formula

8. LOG
Remember on the
calculator the key that
looks like this? :
That only works for one base!
Base 10Better known as the
“common” log.

9. Our calculator has the
base-10 Log
programmed for every
number
Example: 22log
Answer: 1.34
4log3
From base-3, to base-10?
Maybe we can use that
fact to convert

10. Introducing….
The Change of Base Formula
a
x
x
b
b
a
log
log
log 
Now we can solve our logarithm.
What it means:
I can convert the logarithm of a
base I don’t know into a
fraction of 2 logarithms of a
base that I DO know.

11. Let’s use the change of base formula
to convert the log from base-3 to
base-10:
a
x
x
b
b
a
log
log
log The change of
base formula is:
4log3
3log
4log
10
10
Use your calculator
to solve:
262.14log3 

12. We could have made it any other base besides
base 10 , but 10 is more practical and useful:
a
x
x
b
b
a
log
log
log The change of
base formula is:
4log3
Base π :
log 22 4
log 22 3Base 22:
4log3
log π 4
log π 3
Kind of cool, but you can’t really put in most calculators.

13. 40log3
9log2
2
1
log7
2log
9log
10
10
 169.3
357.3
356.
3log
40log

7log
2
1
log


14. Solving Equations using change of base

15. To solve an exponential equation that can’t be solved
using the one to one property, take the log of both
sides.
Solve for X: 532
x

16. Solve for X:
To solve an exponential equation, take the log of both
sides
382
x

17. Solve for X:
To solve an exponential equation, take the log of both
sides
52𝑥−1
4
= 2.66

18. Solve for X:
26.1log x623.log x