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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Exponential & Logarithmic Functions

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Exponential logarithmic functions

Exponential & Logarithmic Functions


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)


Relationships of exponential y b x logarithmic y log b x functions1

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


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


That s all folks

That’s All Folks !


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