13.4.1: Resources and Key Concepts

Math 375: Pre-Calculus · LibreTexts

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    From LibreTexts, under not declared, as the source states it; its equations are written in TeX, which the PDF shows typeset. This page at LibreTexts.

    Key Concepts

    Theorems

    • Domain and Range of the Logarithmic Function \(f(x) = \log_b(x)\): The domain is \((0, \infty)\) and the range is \((-\infty, \infty)\).
    • Properties of the Graphs of Logarithmic Functions \(f(x)=\log_b(x)\) (where \(b>0, b \neq 1\)):
      • One-to-one function.
      • Vertical asymptote: \(x=0\).
      • Domain: \((0, \infty)\).
      • Range: \((-\infty, \infty)\).
      • \(x\)-intercept: \((1,0)\).
      • Secondary key point: \((b,1)\).
      • \(y\)-intercept: none.
      • Increasing if \(b>1\).
      • Decreasing if \(0 < b < 1\).

    Common Mistakes

    • Incorrectly Identifying the Domain of Transformed Logarithmic Functions: The argument of the logarithm must always be positive. For \(f(x) = \log_b(expression)\), one must solve \(expression > 0\). Errors occur if this inequality is set up or solved incorrectly, especially with reflections or multiple transformations.
    • Errors in Horizontal Shifts and Vertical Asymptote Location: Confusing the direction of horizontal shifts (e.g., \(\log_b(x+c)\) shifts left, \(\log_b(x-c)\) shifts right for \(c>0\)) and consequently misplacing the vertical asymptote. The vertical asymptote shifts with the horizontal shift.
    • Order of Transformations: Applying shifts, reflections, and stretches/compressions in the wrong order, particularly when combining horizontal transformations (e.g., in \(h(x) = 2\ln(-\frac{1}{4}x+1)-1\), factoring the argument is key: \(-\frac{1}{4}(x-4)\)).
    • Misinterpreting Stretches vs. Shifts: Confusing the effect of multiplying the function/input (stretch/compression) with adding to the function/input (shift).
    • Incorrectly Identifying the Range of Logarithmic Functions: The range of any untransformed or vertically stretched/compressed/shifted logarithmic function \(y = a\log_b(expression) + d\) is always \((-\infty, \infty)\). Horizontal transformations do not affect the range.
    • Sign Errors with Reflections: Incorrectly applying reflections; \(-f(x)\) reflects across the \(x\)-axis, and \(f(-x)\) reflects across the \(y\)-axis. This also impacts the domain for \(f(-x)\).
    • Graphing Absolute Value of Logarithmic Function: Incorrectly reflecting parts of the graph. Only the portions of the original logarithmic graph that are below the \(x\)-axis should be reflected above the \(x\)-axis. Portions already above remain unchanged.

    This page titled 13.4.1: Resources and Key Concepts was last modified on Tue, 08 Jul 2025 17:47:25 GMT and is shared under a not declared license and was authored, remixed, and/or curated by Roy Simpson.