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11-5 Common Logs and Natural Logs

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11-5 Common Logsand Natural Logs

- Any log that has base 10 is called a common logarithm. We usually do not write the 10.
- We can figure out many common logs easily
- Ex) log 100 =2
- log 1000= 3

- You can apply all properties to common logs. You can use a calculator to help.
- Evaluate:
- 1) log5(2)32) log

Sometimes, we can’t evaluate logs in our heads. Then, we use the change of base formula to help.

Use change of base to evaluate:

3) Log910434) log8 172

Use logarithms to solve

5) 63x = 816) 54z = 73

7) .25 = log16x

Sometimes, we can use our graphing calculators to help us solve logs

8) Solve by graphing 5x-1 = 2x

e

The number e is a famous irrational number, and is one of the most important numbers in mathematics.

2.7182818284590452353602874713527

(we will just use 2.718)

Calculating Interest

Simple InterestContinuously Compounded

A= total, P = initial amount (principal)

r= rate n=times compounded t=time

Sometimes we use e in an exponential function. When we have base e, we use a natural logarithm (ln x) to solve.

Solve

10) 18 = e 3x11) 4.5 > e .031x

In a study at White Oak, scientist found that time t=0, 5% of people have heard a rumor. After 2 days, 25% of students heard that rumor. How long will it take for 80% of White Oak Students to hear that rumor?

13) Convert log6 254 to a natural log, then evaluate.

14) Solve 6.5 = -16.25 lnx

15) Solve 48