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Exponential & Logarithmic Functions

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Exponential & Logarithmic Functions. Table of Contents. Exponential Functions Logarithmic Functions Converting between Exponents and Logarithms Properties of Logarithms Exponential and Logarithmic Equations. General Form of Exponential Function y = b x where b > 1.

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table of contents
Table of Contents
  • Exponential Functions
  • Logarithmic Functions
  • Converting between Exponents and Logarithms
  • Properties of Logarithms
  • Exponential and Logarithmic Equations
general form of exponential function y b x where b 1
General Form of Exponential Function y = b x where b > 1
  • Domain: All reals
  • Range:

y > 0

  • x-intercept: None
  • y-intercept: (0, 1)
general form of exponential function y b x c d where b 1
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)
relationships of exponential y b x logarithmic y log b x functions
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)
converting between exponents logarithms
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
logarithmic abbreviations
Logarithmic Abbreviations
  • log10 x = log x (Common log)
  • loge x = ln x (Natural log)
  • e = 2.71828...
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) = log11 8-1 = – 1 log11 8 = – log11 8

Properties of Logarithms
properties of logarithms shortcuts
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)
examples of logarithms
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

change of base formula
logam

logbm = --------

logab

log712 = log 12

log 7

OR

Change-of-Base Formula
  • log712= ln 12
  • ln 7
exponential logarithmic equations
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.

if your variable is in the exponent
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.
logarithmic equations
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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