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Properties of Logarithms

Properties of Logarithms. Information. What is a logarithm?. The inverse of an exponential is called a logarithm (log). logarithm: if x = a y , then y = log a x. for a positive value of x and for a > 0, a ≠ 1. “log a x ” is read as “log base a of x .”.

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Properties of Logarithms

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  1. Properties of Logarithms

  2. Information

  3. What is a logarithm? The inverse of an exponential is called a logarithm (log). logarithm: if x = ay, then y = logax for a positive value of xand for a > 0, a ≠ 1 “logax” is read as “log base a of x.” The logarithm base 10 of x can be written as just “logx”. The logarithm base e (approximately 2.718) of x is called the natural logarithm and is denoted “lnx”. Rewrite the following as logarithmic equations: a) p = km b) 5 = ex c) 81 = 34 a) m = logk p b) ln5 = x c) 4 = log3 81

  4. Calculating logarithms A scientific calculator can be used to evaluate logarithms base 10 (using the “log” button) and logarithms base e (using the “ln” button). For example, to calculate log4, press “log” followed by “4”. Find the following logarithms to the nearest thousandth: = 0.903 = 2.773 a) log8 d) 2ln4 = 2.773 = 1.099 b) ln16 e) ln 12 – ln 4 = 1.099 = 0.903 c) ln3 f) log 2 + log 4 What do you notice about the results? Can you use theseresults to establish any rules?

  5. Basic rules of logarithms

  6. Taking logarithms Logarithms are the inverse of exponentials. For example, to solve ex = 10, take natural logarithms (base e) of both sides to get: x = ln10. This can now be solved for x. Solve 10x = 15 to the nearest thousandth. 10x = 15 take logs (base 10) of both sides: x = log15 evaluate using a calculator: x = 1.176 Expressions with bases other than 10 or e cannot be evaluated directly using a calculator. To calculate these, the base must be changed to 10 or e.

  7. The change of base rule Expressions with bases other than 10 or e can be rewritten in base 10 or e to be evaluated directly using a calculator. How can logb x = m be rewritten using a logarithm base a? Hint: rewrite it in exponent form and then take logarithms base a. In exponent form, this expression is: bm = x take logs (base a) of both sides: loga (bm) = loga x exponent rule: mloga b = loga x m= loga x / loga b rearrange: change of base rule: logb x = loga x / loga b

  8. Practice changing bases

  9. Logarithmic properties

  10. Proving the logarithmic rules

  11. Logarithmic scales

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