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Logarithms. Logarithms can be very helpful when solving exponential equations , specifically when they do not have the same base. In fact, logarithms ARE exponents. Def : What is a logarithm?

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logarithms
Logarithms
  • Logarithms can be very helpful when solving exponential equations , specifically when they do not have the same base. In fact, logarithms ARE exponents.
  • Def: What is a logarithm?

Given: = a, where b represents the base, x represent the exponent and “a” represents the answer. Both b and x are positive numbers where 1

This can be written using logarithms:

  • Again, b is the base, a is the answer and x is the exponent. This allows us to solve for the variable when it is in the exponent.
  • = a - is called EXPONENTIAL FORM
  • = x - is called LOGARITHMIC FORM
  • If you are asked to concert from exponential form to logarithmic form, you simply substitute in the base, answer and exponent
  • ie. = 16 can be written = 2 try: = 125
slide2
But now what happens, when asked to evaluate a simple logarithm such as .
  • Remember the acronym base, answer, exponent. So, we ask ourselves: “6 raised to what power equals 36?”
  • Since 6 is the base and 36 is the answer, your are trying to find what the exponent is. In this case, the answer is 2 because 6 raised to the second power is 36.
  • Let’s try some: Evaluate: - the answer is 5 since 2 raised to the 5th power is 3
  • Evaluate: - the answer is 3 since 10 raised to the 3rd power is 1000
slide3
You must keep in mind that not all log functions can be done in your head: A few easy ones first
  • 1. Set log = y

2. Change to exponential form

  • 3. Determine if 27 is a power of 3
  • 4. Set exponents equal and solve
  • = y = y
  • = 27
  • y = 3
  • y = -3
determining the domain of a log
Determining the Domain of a Log
  • A log fn. = y is defined as the inverse exponential function: = x
  • So if f(x) = (x) = then (x) =
  • We Know:
  • DOMAIN = RANGE f
  • RANGE = DOMAIN f
  • Thus it follows:

Domain of a LOG = Range of EXPONENTIAL FN = (0 ,

Range of a LOG =Domain of EXPONENTIAL FN =

The Domain of a Log is Positive Real Numbers so the argument of a log fn. Must be > 0

finding the domaing of a log
FINDING THE DOMAING OF A LOG
  • f(x) =
  • y =
  • = x + 3 D: x + 3 > 0

x > -3 D: (-3,

Try: f(x) = g(x) =

h(x) =

slide6
g(x) =
  • y =
  • D: > 0 b/c it’s a fraction must

solve both num.& den.

  • 1 + x > 0 1 – x > 0

x > -1 -x > -1

x < 1

D: (-1,1)

slide7
h(x) =
  • y =
  • = D: > 0
  • - x > 0 x > 0
  • x < 0
  • D: All Reals where x0
change of base
Change of Base

Evaluate

3x = 7

loga M=

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