1. Sec. 5.3 Properties of Logarithms

2. Solve for x log230 = x
2x = 30
30 can’t be broken down to a power of 2.
However, since we know
24 = 16 and 25 = 32 and 30 is between 16 and 32
x is between 4 & 5

3. This is estimated!
To find exact answers use change of base formula or

4. Ex. 1 Use change of base formula
a) log430
b) log214
You can change to any base but to use the calculator you must change the base to 10 or the natural base e
( log key or ln key)

5. Ex. 2 evaluate using ln
a) log430
b) log214

6. Evaluate
1) log74 2) log )log⅓0.015

7. Properties of Logarithms
loga(uv) = logau + logav
(like the property of exponents where you add the exp. if they have the same base.)
2) loga(u/v) = logau – logav
(Comes from the division of exponents)
3) logaun = n logau
These properties are also true for natural logs.

8. Applying these properties is also called expanding the logarithmic expression

9. Ex. 3 Write in terms of ln 2 and ln 3
a) ln 6
b) ln 2/27

10. Ex.
log3(92 • 43)
log2 (8/11)
ln(e3/5)

11. Condensing Logarithmic Expressions
½ log10x +3 log10 (x + 1)
Ex.8
2 ln (x + 2) – ln x

12. Ex. ln (x + 2) – ln 8 log45 + log4x 8 log3 (x – 1) 3 log45 + log48
ln x + ln 3 – ln (x + 2)
5 log6 (x + 2) – 3 log6 x