# Logarithms - PowerPoint PPT Presentation

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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 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

• 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

• 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

• 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

• f(x) =

• y =

• = x + 3 D: x + 3 > 0

x > -3 D: (-3,

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

h(x) =

• 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)

• h(x) =

• y =

• = D: > 0

• - x > 0 x > 0

• x < 0

• D: All Reals where x0

Evaluate

3x = 7

loga M=