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10-4 Common logarithms

10-4 Common logarithms. What you ’ ll learn. Solve exponential equations and inequalities using common logarithms Evaluate logarithmic expressions using the Change of Base Formula. Key words. Common logarithms : logarithms that use 10 as the base

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10-4 Common logarithms

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  1. 10-4 Common logarithms

  2. What you’ll learn • Solve exponential equations and inequalities using common logarithms • Evaluate logarithmic expressions using the Change of Base Formula

  3. Key words • Common logarithms : logarithms that use 10 as the base • Change of Base Formula : allows you to write equivalent logarithmic expressions that have different bases.

  4. Common logarithms • Common logarithms are usually written without the subscript 10 Ex) log10 x = log x, x > 0

  5. Change of Base Formula • For all positive numbers, a, b and n, where a and b doesn’t equal to 1, • log a n = log b n / log b a Ex) log5 = log10 12 / log 10 5

  6. Change of Base Formula example • Express log4 25 in terms of common logarithms. Then approximate its value to four decimal places. Log4 25= log10 25 / log10 4 ≈ 2.3219

  7. Proving the formula • To prove the change of base formula, let log a n = x, ax = n Log b ax = log b n x log b a = log b n x = log b n / log b a Log a n = log b n / log b a

  8. Solving Exponential Equations • Solve 3x = 11 3x = 11 Log 3x = log 11 x log 3 = log 11 x = log 11 / log 3 x ≈ 1.0414 / 0.4771 x ≈ 2.1828

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