Key Equations
General Form for the Transformation of the Parent Logarithmic Function f(x)=logb(x) | f(x)=alogb(x+c)+d |
Key Concepts
- To find the domain of a logarithmic function, set up an inequality showing the argument greater than zero and solve for x.
- The graph of the parent function f(x)=logb(x) has an x-intercept at (1,0), domain (0,∞), range (−∞,∞), vertical asymptote x = 0, and
- if b > 1, the function is increasing.
- if 0 < b < 1, the function is decreasing.
- The equation f(x)=logb(x+c) shifts the parent function y=logb(x) horizontally
- left c units if c > 0.
- right c units if c < 0.
- The equation f(x)=logb(x)+d shifts the parent function y=logb(x) vertically
- up d units if d > 0.
- down d units if d < 0.
- For any constant a > 0, the equation f(x)=alogb(x)
- stretches the parent function y=logb(x) vertically by a factor of a if |a| > 1.
- compresses the parent function y=logb(x) vertically by a factor of a if |a| < 1.
- When the parent function y=logb(x) is multiplied by –1, the result is a reflection about the x-axis. When the input is multiplied by –1, the result is a reflection about the y-axis.
- The equation f(x)=−logb(x) represents a reflection of the parent function about the x-axis.
- The equation f(x)=logb(−x) represents a reflection of the parent function about the y-axis.
- A graphing calculator may be used to approximate solutions to some logarithmic equations.
- All transformations of the logarithmic function can be summarized by the general equation f(x)=alogb(x+c)+d.
- Given an equation with the general form f(x)=alogb(x+c)+d, we can identify the vertical asymptote x = –c for the transformation.
- Using the general equation f(x)=alogb(x+c)+d, we can write the equation of a logarithmic function given its graph.
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- College Algebra. Authored by: Abramson, Jay et al.. Provided by: OpenStax. Located at: http://cnx.org/contents/9b08c294-057f-4201-9f48-5d6ad992740d@5.2. License: CC BY: Attribution. License Terms: Download for free at http://cnx.org/contents/9b08c294-057f-4201-9f48-5d6ad992740d@5.2