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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?

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Logarithms

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  1. 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

  2. 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

  3. 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

  4. 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

  5. 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) =

  6. 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)

  7. h(x) = • y = • = D: > 0 • - x > 0 x > 0 • x < 0 • D: All Reals where x0

  8. Solving Logarithmic Equations

  9. Change of Base Evaluate 3x = 7 loga M=

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