Logarithm Properties Notes
Product:
Quotient:
Power:
Use the
properties to
expand each
logarithm.
1. 𝑙𝑜𝑔3 5𝑥
8
2. 𝑙𝑜𝑔2 𝑟 3
Common Log:
4. 𝑙𝑜𝑔5 √𝑤𝑥 6
3. 𝑙𝑛 𝑤
Log with no base is
considered base 10
Natural Log:
𝑚
6. 𝑙𝑜𝑔𝑥 𝑚4 𝑛
5.𝑙𝑛 𝑛3
𝑎 5
7. 𝑙𝑜𝑔 (𝑏 )
𝑚
8. 𝑙𝑜𝑔2 √𝑛3
ln is called the natural log
*base e
Use the
properties to
condense
each
expression to
a single
logarithm
9. 𝑙𝑜𝑔𝑎 3 + 𝑙𝑜𝑔𝑎 4
10. ln 7 − ln 5
11. 3 𝑙𝑜𝑔𝑏 2
12.
After you
condense,
LOOK to see
if you can
evaluate the
logarithm
and simplify
𝑙𝑜𝑔𝑎 36
2
13. ln 6 + ln 5 − ln 2
14.
15. 2 𝑙𝑜𝑔3 6 − 𝑙𝑜𝑔3 4
16. 2 log 5 + log 4
17. 𝑙𝑜𝑔4 40 − 𝑙𝑜𝑔4 5
18. 𝑙𝑜𝑔4 3 − 𝑙𝑜𝑔4 48
2 𝑙𝑜𝑔𝑑 𝑤
3
Given that 𝒍𝒐𝒈𝟐 𝟑 = 𝟏. 𝟓𝟗 and 𝒍𝒐𝒈𝟐 𝟓 = 𝟐. 𝟑𝟐 (accurate to 2 decimal places), find the following.
Steps:
1. Expand the functions
2. Evaluate each log
3. Simplify
5
19. 𝑙𝑜𝑔2 9
20. 𝑙𝑜𝑔2 3
22. 𝑙𝑜𝑔2 15
21. 𝑙𝑜𝑔2 125
23. 𝑙𝑜𝑔2
15
9
Algebra 2 PreAP
Properties of Logarithms
Expand each logarithm.
1. log 3 𝑚 6 𝑛3
4. log 𝑥 3(𝑔2 ℎ)
2. ln(𝑎𝑏)4
3. log 2 𝑏√𝑐
𝑙
6. ln 𝑗𝑘
𝑥4
5. log 5 𝑦3
Condense each expression into a single logarithm.
7. log 𝑎 𝑥 − 4 log 𝑎 𝑦
10.
𝑙𝑜𝑔𝑎 8
3
1
1
8. 4 ln 𝐴 + 2 ln 𝐵
9. 3 𝑙𝑜𝑔𝑏 2 − 3 log 𝑏 𝑟
11. (hint: evaluate after condensing) 2 log 2 4 − log 2 8
If log 2 = .3 and log 5 = .7, then find the following using your properties of logarithms
12. log 10
13. log (2/5)
14. log 20
15. log (5/2)
16. log 4
17. log 125
Algebra 2 PreAP
Properties of Logarithms
Expand each logarithm.
1. log 3 𝑚 6 𝑛3
4. log 𝑥 3(𝑔2 ℎ)
2. ln(𝑎𝑏)4
5. log 5
3. log 2 𝑏√𝑐
𝑙
6. ln
𝑥4
𝑗𝑘
𝑦3
Condense each expression into a single logarithm.
7. log 𝑎 𝑥 − 4 log 𝑎 𝑦
𝑙𝑜𝑔 8
1
1
8. 4 ln 𝐴 + 2 ln 𝐵
9. 3 𝑙𝑜𝑔𝑏 2 − 3 log 𝑏 𝑟
11. (hint: evaluate after condensing) 2 log 2 4 − log 2 8
𝑎
10.
3
If log 2 = .3 and log 5 = .7, then find the following using your properties of logarithms
12. log 10
13. log (2/5)
14. log 20
15. log (5/2)
16. log 4
17. log 125
Algebra 2 PreAP
Properties of Logarithms
Expand each logarithm.
1. log 3 𝑚 6 𝑛3
4. log 𝑥 3(𝑔2 ℎ)
2. ln(𝑎𝑏)4
3. log 2 𝑏√𝑐
𝑙
6. ln 𝑗𝑘
𝑥4
5. log 5 𝑦3
Condense each expression into a single logarithm.
7. log 𝑎 𝑥 − 4 log 𝑎 𝑦
10.
𝑙𝑜𝑔𝑎 8
3
1
1
8. 4 ln 𝐴 + 2 ln 𝐵
9. 3 𝑙𝑜𝑔𝑏 2 − 3 log 𝑏 𝑟
11. (hint: evaluate after condensing) 2 log 2 4 − log 2 8
If log 2 = .3 and log 5 = .7, then find the following using your properties of logarithms
12. log 10
13. log (2/5)
14. log 20
15. log (5/2)
16. log 4
17. log 125