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Exponential & Logarithmic FunctionsPowerPoint Presentation

Exponential & Logarithmic Functions

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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

- Domain: All reals
- Range:
y > 0

- x-intercept: None
- y-intercept: (0, 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 = 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 = bx) & Logarithmic (y = logbx) Functions

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 & LogarithmsLogarithmic Abbreviations

- log10 x = log x (Common log)
- loge x = ln x (Natural log)
- e = 2.71828...

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 LogarithmsProperties 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

- 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

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…..

- 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

- 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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