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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.

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Exponential & Logarithmic Functions

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  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 (y = logbx) Functions

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

  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 !

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