Logarithms
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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

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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


Logarithms

  • 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


Logarithms

  • 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) =


Logarithms

  • 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)


Logarithms

  • h(x) =

  • y =

  • = D: > 0

  • - x > 0 x > 0

  • x < 0

  • D: All Reals where x0


Solving logarithmic equations

Solving Logarithmic Equations


Change of base

Change of Base

Evaluate

3x = 7

loga M=


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