{"id":274,"date":"2023-06-21T13:22:51","date_gmt":"2023-06-21T13:22:51","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/summary-graphs-of-logarithmic-functions\/"},"modified":"2023-06-21T13:22:51","modified_gmt":"2023-06-21T13:22:51","slug":"summary-graphs-of-logarithmic-functions","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/gsu-collegealgebra\/chapter\/summary-graphs-of-logarithmic-functions\/","title":{"raw":"Summary: Graphs of Logarithmic Functions","rendered":"Summary: Graphs of Logarithmic Functions"},"content":{"raw":"\n\n
Key Equations<\/h2>\n
\n\n
\n
General Form for the Transformation of the Parent Logarithmic Function [latex]\\text{ }f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/td>\n
To find the domain of a logarithmic function, set up an inequality showing the argument greater than zero and solve for x<\/em>.<\/li>\n \t
The graph of the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] has an x-<\/em>intercept at [latex]\\left(1,0\\right)[\/latex], domain [latex]\\left(0,\\infty \\right)[\/latex], range [latex]\\left(-\\infty ,\\infty \\right)[\/latex], vertical asymptote x <\/em>= 0, and\n
\n \t
if b <\/em>> 1, the function is increasing.<\/li>\n \t
if 0 < b <\/em>< 1, the function is decreasing.<\/li>\n<\/ul>\n<\/li>\n \t
The equation [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x+c\\right)[\/latex] shifts the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] horizontally\n
\n \t
left c<\/em> units if c <\/em>> 0.<\/li>\n \t
right c<\/em> units if c <\/em>< 0.<\/li>\n<\/ul>\n<\/li>\n \t
The equation [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)+d[\/latex] shifts the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] vertically\n
\n \t
up d<\/em> units if d <\/em>> 0.<\/li>\n \t
down d<\/em> units if d <\/em>< 0.<\/li>\n<\/ul>\n<\/li>\n \t
For any constant a <\/em>> 0, the equation [latex]f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x\\right)[\/latex]\n
\n \t
stretches the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] vertically by a factor of a<\/em> if |a<\/em>| > 1.<\/li>\n \t
compresses the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] vertically by a factor of a<\/em> if |a<\/em>| < 1.<\/li>\n<\/ul>\n<\/li>\n \t
When the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] is multiplied by \u20131, the result is a reflection about the x<\/em>-axis. When the input is multiplied by \u20131, the result is a reflection about the y<\/em>-axis.\n
\n \t
The equation [latex]f\\left(x\\right)=-{\\mathrm{log}}_{b}\\left(x\\right)[\/latex] represents a reflection of the parent function about the x-<\/em>axis.<\/li>\n \t
The equation [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(-x\\right)[\/latex] represents a reflection of the parent function about the y-<\/em>axis.<\/li>\n<\/ul>\n
\n \t
A graphing calculator may be used to approximate solutions to some logarithmic equations.<\/li>\n<\/ul>\n<\/li>\n \t
All transformations of the logarithmic function can be summarized by the general equation [latex] f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x+c\\right)+d[\/latex].<\/li>\n \t
Given an equation with the general form [latex] f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x+c\\right)+d[\/latex], we can identify the vertical asymptote x <\/em>= \u2013c for the transformation.<\/li>\n \t
Using the general equation [latex]f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x+c\\right)+d[\/latex], we can write the equation of a logarithmic function given its graph.<\/li>\n<\/ul>\n\n","rendered":"
Key Equations<\/h2>\n
\n\n
\n
General Form for the Transformation of the Parent Logarithmic Function [latex]\\text{ }f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/td>\n
To find the domain of a logarithmic function, set up an inequality showing the argument greater than zero and solve for x<\/em>.<\/li>\n
The graph of the parent function [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] has an x-<\/em>intercept at [latex]\\left(1,0\\right)[\/latex], domain [latex]\\left(0,\\infty \\right)[\/latex], range [latex]\\left(-\\infty ,\\infty \\right)[\/latex], vertical asymptote x <\/em>= 0, and\n
\n
if b <\/em>> 1, the function is increasing.<\/li>\n
if 0 < b <\/em>< 1, the function is decreasing.<\/li>\n<\/ul>\n<\/li>\n
The equation [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x+c\\right)[\/latex] shifts the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] horizontally\n
\n
left c<\/em> units if c <\/em>> 0.<\/li>\n
right c<\/em> units if c <\/em>< 0.<\/li>\n<\/ul>\n<\/li>\n
The equation [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(x\\right)+d[\/latex] shifts the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] vertically\n
\n
up d<\/em> units if d <\/em>> 0.<\/li>\n
down d<\/em> units if d <\/em>< 0.<\/li>\n<\/ul>\n<\/li>\n
For any constant a <\/em>> 0, the equation [latex]f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x\\right)[\/latex]\n
\n
stretches the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] vertically by a factor of a<\/em> if |a<\/em>| > 1.<\/li>\n
compresses the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] vertically by a factor of a<\/em> if |a<\/em>| < 1.<\/li>\n<\/ul>\n<\/li>\n
When the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] is multiplied by \u20131, the result is a reflection about the x<\/em>-axis. When the input is multiplied by \u20131, the result is a reflection about the y<\/em>-axis.\n
\n
The equation [latex]f\\left(x\\right)=-{\\mathrm{log}}_{b}\\left(x\\right)[\/latex] represents a reflection of the parent function about the x-<\/em>axis.<\/li>\n
The equation [latex]f\\left(x\\right)={\\mathrm{log}}_{b}\\left(-x\\right)[\/latex] represents a reflection of the parent function about the y-<\/em>axis.<\/li>\n<\/ul>\n
\n
A graphing calculator may be used to approximate solutions to some logarithmic equations.<\/li>\n<\/ul>\n<\/li>\n
All transformations of the logarithmic function can be summarized by the general equation [latex]f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x+c\\right)+d[\/latex].<\/li>\n
Given an equation with the general form [latex]f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x+c\\right)+d[\/latex], we can identify the vertical asymptote x <\/em>= \u2013c for the transformation.<\/li>\n
Using the general equation [latex]f\\left(x\\right)=a{\\mathrm{log}}_{b}\\left(x+c\\right)+d[\/latex], we can write the equation of a logarithmic function given its graph.<\/li>\n<\/ul>\n\n\t\t\t \n\t\t\t