1 / 16

Exponential & Logarithmic Functions

Exponential & Logarithmic Functions. Dr. Carol A. Marinas. Table of Contents. Exponential Functions Logarithmic Functions Converting between Exponents and Logarithms Properties of Logarithms Exponential and Logarithmic Equations.

iniko
Download Presentation

Exponential & Logarithmic Functions

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Exponential & Logarithmic Functions Dr. Carol A. Marinas

  2. Table of Contents • Exponential Functions • Logarithmic Functions • Converting between Exponents and Logarithms • Properties of Logarithms • Exponential and Logarithmic Equations

  3. General Form of Exponential Function y = b x where b > 1 • Domain: All reals • Range: y > 0 • x-intercept: None • y-intercept: (0, 1)

  4. General Form of Exponential Functiony = b (x + c) + dwhere b > 1 • c moves graph left or right (opposite way) • d move graph up or down (expected way) • So y=3(x+2) + 3 moves the graph 2 units to the left and 3 units up • (0, 1) to (– 2, 4)

  5. Relationships of Exponential (y = bx) & Logarithmic (y = logbx) Functions • y = logbx is the inverse of y = bx • Domain: x > 0 • Range: All Reals • x-intercept: (1, 0) • y-intercept: None • y = bx • Domain: All Reals • Range: y > 0 • x-intercept: None • y-intercept: (0, 1)

  6. Relationships of Exponential(y = bx) Logarithmic Functions (y = logbx)

  7. BASEEXPONENT = POWER 42 = 16 4 is the base. 2 is the exponent. 16 is the power. Converting between Exponents & Logarithms As a logarithm logBASEPOWER=EXPONENT log 4 16 = 2

  8. Logarithmic Abbreviations • log10 x = log x (Common log) • loge x = ln x (Natural log) • e = 2.71828...

  9. logb(MN)= logbM + logbN Ex: log4(15)= log45 + log43 logb(M/N)= logbM – logbN Ex: log3(50/2)= log350 – log32 logbMr = r logbM Ex: log7 103 = 3 log7 10 logb(1/M) = logbM-1= –1 logbM = – logbM log11 (1/8) = log11 8-1 = – 1 log11 8 = – log11 8 Properties of Logarithms

  10. Properties of Logarithms (Shortcuts) • logb1 = 0 (because b0 = 1) • logbb = 1 (because b1 = b) • logbbr = r (because br = br) • blog b M = M (because logbM= logbM)

  11. Examples of Logarithms • Simplify log 7 + log 4 – log 2 = log 7*4 = log 14 2 • Simplify ln e2= 2 ln e = 2 logee = 2 * 1 = 2 • Simplify e 4 ln 3 - 3 ln 4= e ln 34 - ln 43 = e ln 81/64 = e loge81/64 = 81/64

  12. logam logbm = -------- logab log712 = log 12 log 7 OR Change-of-Base Formula • log712= ln 12 ln 7

  13. Exponential & Logarithmic Equations • If logb m = logb n, then m = n. If log6 2x = log6(x + 3), then 2x = x + 3 and x = 3. • If bm = bn, then m = n. If 51-x = 5-2x, then 1 – x = – 2x and x = – 1.

  14. If your variable is in the exponent….. • Isolate the base-exponent term. • Write as a log. Solve for the variable. • Example: 4x+3 = 7 • log4 7 = x + 3 and – 3 + log4 7 = x OR with change of bases: x = – 3 + log 7 log 4 • Another method is to take the LOG of both sides.

  15. Logarithmic Equations • Isolate to a single log term. • Convert to an exponent. • Solve equation. • Example: log x + log (x – 15) = 2 • log x(x – 15) = 2 so 102 = x (x – 15) and 100 = x2 – 15x and 0 = x2 – 15x – 100 So 0 = (x – 20) (x + 5) so x = 20 or – 5

  16. That’s All Folks !

More Related