3.4 Exponential & Logarithmic Equations

1. 3.4Exponential & Logarithmic Equations JMerrill, 2010

2. Quick Review of 3.3 • Properties of Logs

3. Rules of LogarithmsIf M and N are positive real numbers and b is ≠ 1: • The Product Rule: • logbMN = logbM + logbN (The logarithm of a product is the sum of the logarithms) • Example: log (10x) = log10 + log x • You do: log7(1000x) = • log71000 + log7x

4. Rules of LogarithmsIf M and N are positive real numbers and b ≠ 1: • The Quotient Rule (The logarithm of a quotient is the difference of the logs) • Example: • You do:

5. Rules of LogarithmsIf M and N are positive real numbers, b ≠ 1, and p is any real number: • The Power Rule: • logbMp = p logbM (The log of a number with an exponent is the product of the exponent and the log of that number) • Example: log x2 = 2 log x • Example: log574 = 4 log57 • You do: log359 • Challenge: = 9 log35

6. Condensing • Sometimes, we need to condense before we can solve: Product Rule Power Rule Quotient Rule

7. Condensing • Condense:

8. Using the Rules to Condense • Ex: • You Do:

9. Bases • We don’t really use other bases anymore, but since logs are often written in other bases, we must change to base 10 in order to use our calculators.

10. Change of Base Formula • Example log58= • This is also how you graph in another base. Enter y1=log(8)/log(5). Remember, you don’t have to enter the base when you’re in base 10! Parentheses are vital! The log key opens the ( ), you must close it!

11. 3.4Solving Exponents & Logs

12. Solving Guidelines Get both parts to the same base If you have a variable in the exponent position, take the log of both sides. Take the ln if you’re using e, take the log if using common logs. • Original Rewritten Solution • 2x = 32 2x = 25 x = 5 • lnx – ln3 = 0 lnx = ln3 x = 3 • (1/3)x = 9 3-x = 32 x = -2 • ex = 7 lnex = ln7 x = ln7 • logx =-1 10logx = 10-1 x = 10-1 = 1/10 Solve like normal Get both parts to the same base If you have a log on one side, exponentiate both sides

13. Solving • Getting all the numbers to the same base. • Example:

14. Solving • Clear the exponent:

15. Solving Exponentials • Exponentiating: • ex = 72 • lnex = ln72 • x = ln72 ≈ 4.277 • You should always check your answers by plugging them back in. Sometimes they don’t work because you can’t take the log of a negative number.

16. Solving Exponentials • 3(2x) = 42 • 2x = 14 • log22x = log214 • x = log214 • x = log14/log2 ≈ 3.807

17. Solving Exponentials • 4e2x – 3 = 2 • 4e2x = 5 • e2x = 5/4 • lne2x = ln 5/4 • 2x = ln 5/4 • x = ½ ln 5/4 ≈ 0.112

18. Solving Exponentials • 2(32t-5) – 4 = 11 • 2(32t-5) = 15 • (32t-5)= 15/2 • log3(32t-5) = log3 15/2 • 2t – 5 = log3 15/2 • 2t = 5 + log3 7.5 • t = 5/2 + ½ log3 7.5 • t ≈ 3.417

19. Solving Exponentials • e2x – 3ex + 2 = 0 • No like terms—kinda look quadratic? • (ex – 2)(ex – 1) = 0 • Set each factor = 0 and solve • (ex – 2) = 0 • ex = 2 • lnex = ln2 • x = ln2 ≈ 0.693 (ex – 1) = 0 ex = 1 lnex = ln 1 x = 0

20. Solving Logarithms • Exponentiating with the natural log • lnx = 2 • elnx = e2 • x = e2≈ 7.389

21. Solving Logarithms • log3(5x - 1) = log3(x + 7) • 5x – 1 = x + 7 • 4x = 8 • x = 2

22. Solving Logs – Last Time • 5 + 2lnx = 4 • 2lnx = -1 • lnx = - ½ • elnx = e - ½ • x = e - ½ • x ≈ 0.607

23. Interest Compounded Continuously • If interest is compounded “all the time” (MUST use the word continuously), we use the formula where P0 is the initial principle (initial amount)

24. If you invest $1.00 at a 7% annual rate that is compounded continuously, how much will you have in 4 years? • You will have a whopping $1.32 in 4 years!