1. Lesson 3.3 Properties of Logarithms
Essential Question: How do you rewrite logarithmic expressions to simplify or evaluate them?

2. Before we start…
Evaluate
log

3. What is an logarithmic function?
Non-linear transcendental function
Inverse of exponential function

4. Change of Base
Most calculators have only two types of log keys, one for common logarithms (base 10) and one for natural logarithms (base e).
Although common logs and natural logs are the most frequently used, you may occasionally need to evaluate logarithms to other bases. To do this, you can use the following change-of-base formula.

5. Change-of-Base Formula
Let a, b, and x be positive real numbers such that a ≠ 1 and b ≠ 1. Then log 𝑎 𝑥 can be converted to a different base using any of the following formulas.
Base b
Base 10
Base e
log 𝑎 𝑥 = log 𝑏 𝑥 log 𝑏 𝑎
log 𝑎 𝑥 = log 10 𝑥 log 10 𝑎
log 𝑎 𝑥 = ln 𝑥 ln 𝑎

6. Evaluate each of the following using the change-of-base formula with common logs. Approximate to four decimal places. log 3 16 log 4 25

7. Evaluate each of the following using the change-of-base formula with common logs. Approximate to four decimal places. log 2 12 log 5 22

8. Evaluate each of the following using the change-of-base formula with natural logs. Approximate to four decimal places. log 3 16 log 4 25

9. Evaluate each of the following using the change-of-base formula with natural logs. Approximate to four decimal places. log 2 12 log 5 22

10. Properties of Logarithms
Let a be a positive real number such that 𝑎≠1, and let n be a real number. If u and v are positive real numbers, then the following properties are true.
Logarithm with Base a
Natural Logarithm
1. Product Property:
log 𝑎 𝑢𝑣 = log 𝑎 𝑢 + log 𝑎 𝑣
ln 𝑢𝑣 = ln 𝑢 + ln 𝑣
2. Quotient Property:
log 𝑎 𝑢 𝑣 = log 𝑎 𝑢 − log 𝑎 𝑣
ln 𝑢 𝑣 = ln 𝑢 − ln 𝑣
3. Power Property:
log 𝑎 𝑢 𝑛 =𝑛 log 𝑎 𝑢
ln 𝑢 𝑛 =𝑛 ln 𝑢

11. Write each logarithm in terms of ln 2 and ln 5 .
ln ln 5 32

12. Write each logarithm in terms of ln 2 and ln 3 .
ln ln 2 27

13. Use the properties of logarithms to verify that log 7 5 7 = 1 5 .

14. Expand each logarithmic expression. log 3 𝑥 2 𝑦

15. Expand each logarithmic expression. log 4 5 𝑥 3 𝑦

16. Expand each logarithmic expression.
ln 4𝑥+1 8

17. Condense each logarithmic expression.
1 3 log 𝑥 +5 log 𝑥−3

18. Condense each logarithmic expression.
4 ln 𝑥−4 −2 ln 𝑥

19. Condense each logarithmic expression.
log 3 𝑥 + log 3 𝑥−2

20. Condense each logarithmic expression.
log 2 𝑥 + log 2 𝑥−4

21. The table shows the mean distance x from the sun and the period y (the time it takes a planet to orbit the sun) for each of the six planets that are closest to the sun. In the table, the mean distance is given in astronomical units (where the Earth’s mean distance is defined as 1.0), and the period is given in years.

22. A pebble is dropped into a calm pond, causing ripples in the form of concentric circles. The table below gives the radius r and the area A of the outer ripple in feet. Find an equation that expresses A as a function of r.
Radius, r
Area, A
0.600
1.131
1.200
4.524
1.800
10.179
2.400
18.096
3.000
28.274
3.600
40.715

23. How do you rewrite logarithmic expressions to simplify or evaluate them?

24. Ticket Out the Door
Expand the expression. ln 3𝑥−5 7