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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 - PowerPoint PPT Presentation
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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
Change of Base Formula : allows you to write equivalent logarithmic expressions that have different bases.
Common logarithms
Common logarithms are usually written without the subscript 10
Ex) log10 x = log x, x > 0
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 aEx) log5 = log10 12 / log 10 5
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
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
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