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Logarithms

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- 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

- 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

- 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=