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**Exponential & Logarithmic Functions**

Dr. Carol A. Marinas

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**Table of Contents Exponential Functions Logarithmic Functions**

Converting between Exponents and Logarithms Properties of Logarithms Exponential and Logarithmic Equations

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**General Form of Exponential Function y = b x where b > 1**

Domain: All reals Range: y > 0 x-intercept: None y-intercept: (0, 1)

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**General Form of Exponential Function y = b (x + c) + d where 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)

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**Relationships of Exponential (y = bx) & Logarithmic (y = logbx) Functions**

Domain: All Reals Range: y > 0 x-intercept: None y-intercept: (0, 1) y = logbx is the inverse of y = bx Domain: x > 0 Range: All Reals x-intercept: (1, 0) y-intercept: None

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**Relationships of Exponential (y = bx) & Logarithmic (y = logbx) Functions**

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**Converting between Exponents & Logarithms**

BASEEXPONENT = POWER 42 = 16 4 is the base. 2 is the exponent is the power. As a logarithm, logBASEPOWER=EXPONENT log 4 16 = 2

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**Logarithmic Abbreviations**

log10 x = log x (Common log) loge x = ln x (Natural log) e =

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**Properties of Logarithms**

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) = log = – 1 log11 8 = – log11 8

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**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)

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**Examples of Logarithms**

Simplify log 7 + log 4 – log 2 = log 7*4 = log Simplify ln e2 = 2 ln e = 2 logee = 2 * 1 = 2 Simplify e 4 ln ln 4 = e ln 34 - ln 43 = e ln 81/64 = e loge 81/64 = 81/64

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**Change-of-Base Formula**

logam logbm = logab log712 = log 12 log 7 OR log712 = ln 12 ln 7

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**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.

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**If your variable is in the exponent…..**

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

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**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

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That’s All Folks !

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