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This guide explores essential properties of logarithms, including the change of base formula and how to evaluate logarithmic expressions. It covers key identities such as the product, quotient, and power properties of logarithms. It explains the derivations using natural logarithms and illustrates how to express logarithmic functions in terms of `ln(2)` and `ln(3)`. Practical exercises and examples provide a solid understanding, allowing you to confidently expand and condense logarithmic expressions.
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Properties of Logarithms Section 3.3
3.3 PROPERTIES OF LOGARITHMS Change of base formula: loga x = logb x / logb a = log x / log a = ln x / ln a Evaluate log9 1043. log10 1043 / log10 9 log 1043 / log 9 3.1630 Evaluate using ln: log9 1043. ln 1043 / ln 9 3.1630 Same!
Properties of Logarithms: loga (uv) = loga u + loga v ln (uv) = ln u + ln v loga u/v = loga u – loga v ln u/v = ln u – ln v loga un = n loga u ln un = n ln u
Write each log in terms of ln 2 & ln 3: • ln 6 ln (2 · 3) ln 2 + ln 3 • ln 2/27 ln 2 – ln 27 ln 2 – ln 33 ln 2 – 3 ln 3
Verify that – log10 1/100 = log10 100 -log10 100-1 = -(-1) log10 100 = log10 100 = Expand each expression: • log4 5x3y log4 5 + log4 x3 + log4 y log4 5 + 3 log4 x + log4 y
ln √3x – 5 7 ln (3x-5)1/2 7 ln (3x – 5)1/2 – ln 7 ½ ln (3x – 5) – ln 7 Condense each expression: • ½ log10 x + 3 log10 (x + 1) log x1/2 + log (x + 1)3 log [ √ x (x + 1)3]
2 ln (x + 2) – ln x ln (x + 2)2 – ln x ln (x + 2)2 2 • 1/3 [log2 x + log2 (x – 4)] 1/3 {log2 [x(x – 4)]} log2 [x(x – 4)]1/3 log23√ x(x – 4)
Homework • Page 211-213 10-20 even, 35, 37-55 odd, 72, 74
