1. Expanding and Condensing Logarithms

2. Common Logarithm log10 = log
A common logarithm is a logarithm that is base 10.
When a logarithm is base 10, we don’t write the base.
log10 = log
We like base 10 because we can evaluate it in our calculator. (Use the LOG button.)

3. Evaluate with a calculator
= 1
21) log10 10 =
22) 2 log10 2.5
no solution
23) log10 (-2)
Remember this means 10? = -2

4. More Properties of Logarithms
If loga x = loga y
then x = y

5. Product Property

6. Ex. Express as a sum of logarithms.
1) loga MN
= loga M + loga N
2) logb AT
= logb A + logb T
3) log MATH
= log M + log A + log T + log H

7. Ex. Express as a single logarithm
4) log log5 3
= log5 (19•3)
5) log C + log A + log B + log I + log N
= log CABIN

8. Ex. Express as a sum of logarithms, then simplify.
= log2 4 + log216
= 2 + 4
= 6

9. Ex. 7 Use log53 = 0.683 and log57 = 1.209 to approximate…
= log5 3 + log5 7 =
= 1.892

10. Quotient Property

11. Ex. Express as the difference of logs8) 9)

12. Ex. 10 Use log53 = 0.683 and log57 = 1.209 to approximate…
= log5 3 – log5 7
= – 1.209 =

13. Power Property

14. Ex. Express as a product.
= -5 • logb9
11)
12)

15. Ex. 13 Use log53 = 0.683 and log57 = 1.209 to approximate…
= 2(1.209)
= 2.418

16. Ex. 14 Expand log105x3y
= log105
+ log10x3
+ log10y
= log105
+ 3 log10x
+ log10y

17. Ex. 15 Expand Simplify the division. Simplify the multiplication of 4
Change the radical sign to an exponent
Express the exponent as a product

18. Ex. Condense.
16)
17)

19. Ex 18 Condense Express all products as exponents
Change the fractional exponent to a radical sign.
Simplify the subtraction.
Simplify the addition.

20. Properties of Logarithms
because a0 = 1
logaa = 1
because a1 = a
logaax = x
If loga x= loga y
then x = y
Product Property
Quotient Property
Power Property
Change-of-Base

21. Warning!! Be careful!!