11-5 Common Logs and Natural Logs

Any log that has base 10 is called a common logarithm Any log that has base 10 is called a common logarithm. We usually do not write the 10. For example, log104 =log4

The log button on the calculator ONLY does common logs The log button on the calculator ONLY does common logs. (all are base 10). Use a calculator to find: 1) log 100 2) log 60

For every other base besides 10, we have to use a formula to evaluate For every other base besides 10, we have to use a formula to evaluate. Use change of base to evaluate: 3) Log91043 4) log8 172

Use logarithms to solve 5) 63x = 81 6) 54z = 73 7) Use logarithms to solve 5) 63x = 81 6) 54z = 73 7) .25 = log16x 8) 6x-2=4x

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 Interest Continuously Compounded 𝐴= 𝑃(1+ 𝑟 𝑛 ) 𝑛𝑡 𝐴=𝑃 𝑒 𝑟𝑡 A= total, P = initial amount (principal) r= rate n=times compounded t=time Find the total balance in an account if $4000 was invested at 3% compounded continuously for 4 years.

Sometimes we use e in an exponential function 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 3x 11) 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) Solve using ln. 32x = 7x-1 14) Solve 6. 5 = -16 13) Solve using ln. 32x = 7x-1 14) Solve 6.5 = -16.25 lnx 15) Use a calc to solve 6 𝑥 2 −2 <48