2.07 ExpLog - Logarithmic Base and Transformed

University:
High School
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High School Physics
Academic year:

2024
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2.07 LOGARITHMIC FUNCTIONS: BASIC & TRANSFORMED EXPLORING THE BASICS 𝒇 𝒙 = 𝒍𝒐𝒈𝟐 𝒙 𝒙 -2 -½ -¼ 0 ¼ ½ 1 2 4 𝒇(𝒙) 𝒈 𝒙 = 𝒍𝒐𝒈 𝒙 -2 -½ -¼ 0 ¼ ½ 1 2 4 𝟏 𝟐 𝒙 𝒈(𝒙) SOME CONNECTIONS Through this exploration and from studying Exponentials we know that : • BASE ≠ NEGATIVE • Base ≠ {𝟎, 𝟏} SOME CONNECTIONS In Exponentials we know: • Power ≠ NEGATIVE • Power ≠ 𝟎 So in Logs… 𝒙 > 𝟎 VERTICAL Asymptote at 𝒙 = 𝟎 THE BASIC LOG CURVE IF 𝒄 > 𝟏 IF 𝟎 < 𝒄 < 𝟏 PROPERTIES OF BASIC CURVES IF 𝒄 > 𝟏 𝒅𝒐𝒎 𝒇 = ℝ ∗ + IF 𝟎 < 𝒄 < 𝟏 𝒓𝒂𝒏 𝒇 = ℝ Intercepts 𝒇 𝟎 𝑫𝑵𝑬 𝒇 𝒙 = 𝟎 𝒘𝒉𝒆𝒏 𝒙 = 𝟏 Variation 𝒇↗ 𝒇↘ Extremes 𝑫𝑵𝑬 𝒂𝒔 𝒓𝒂𝒏 𝒇 = ℝ Sign 𝒇 ≥ 𝟎 𝒐𝒗𝒆𝒓 [𝟏, ∞[ 𝒇 ≤ 𝟎 𝒐𝒗𝒆𝒓 ]𝟎, 𝟏] 𝒇 ≥ 𝟎 𝒐𝒗𝒆𝒓 ]𝟎, 𝟏] 𝒇 ≤ 𝟎 𝒐𝒗𝒆𝒓 [𝟏, ∞[ BOP 𝟏 , −𝟏 𝒄 𝟏, 𝟎 (𝒄, 𝟏) TRANSFORMING 𝒇 𝒙 = 𝒂 log 𝒄 (𝒃 𝒙 − 𝒉 ) + 𝒌 PARAMETERS AND THEIR ROLES If 𝒄 > 𝟏 – The higher the value of the base, the tighter the curve sits to the 𝒙-axis If 𝟎 < 𝒄 < 𝟏 – As 𝒄 approaches zero, the tighter the curve sits to the 𝒙-axis **NOTE** No change to the zero PARAMETERS AND THEIR ROLES As “a” increases, the curve steepens As “a” approaches zero, the tighter the curve sits to the 𝒙-axis If 𝒂 < 𝟎 – x-reflection makes graph look like 0 𝟎 → 𝒅𝒐𝒎 = ]𝒉, ∞[ IF: 𝒃 < 𝟎 → 𝒅𝒐𝒎 = ] − ∞, 𝒉[ Range ALWAYS ℝ Initial Value Only exists if 𝒃 and 𝒉 are OPPOSITE signs and 𝒉 ≠ 𝟎 Variation 𝑐>1 0<𝑐<1 𝑓↗ 𝑎, 𝑏 SAME 𝑎, 𝑏 OPP 𝑓↘ 𝑎, 𝑏 OPP 𝑎, 𝑏 SAME HOMEWORK Visions 1 P184: 5, 9, 11