Before working with graphs, we will take a look at the domain (the set of input values) for which the logarithmic function is defined.<\/p>\r\n
Recall that the exponential function is defined as [latex]y={b}^{x}[\/latex] for any real number x<\/em>\u00a0and constant [latex]b>0[\/latex], [latex]b\\ne 1[\/latex], where<\/p>\r\n\r\n In the last section we learned that the logarithmic function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] is the inverse of the exponential function [latex]y={b}^{x}[\/latex]. So, as inverse functions:<\/p>\r\n\r\n Transformations of the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] behave similarly to those of other functions. Just as with other parent functions, we can apply the four types of transformations\u2014shifts, stretches, compressions, and reflections\u2014to the parent function without loss of shape.<\/p>\r\n In Graphs of Exponential Functions<\/a> we saw that certain transformations can change the range<\/em> of [latex]y={b}^{x}[\/latex]. Similarly, applying transformations to the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] can change the domain<\/em>. When finding the domain of a logarithmic function, therefore, it is important to remember that the domain consists only of positive real numbers<\/em>. That is, the argument of the logarithmic function must be greater than zero.<\/p>\r\n For example, consider [latex]f\\left(x\\right)={\\mathrm{log}}_{4}\\left(2x - 3\\right)[\/latex]. This function is defined for any values of x<\/em>\u00a0such that the argument, in this case [latex]2x - 3[\/latex], is greater than zero. To find the domain, we set up an inequality and solve for\u00a0x<\/em>:<\/p>\r\n\r\n In interval notation, the domain of [latex]f\\left(x\\right)={\\mathrm{log}}_{4}\\left(2x - 3\\right)[\/latex] is [latex]\\left(1.5,\\infty \\right)[\/latex].<\/p>\r\n\r\n What is the domain of [latex]f\\left(x\\right)={\\mathrm{log}}_{2}\\left(x+3\\right)[\/latex]?<\/p>\r\n\r\n<\/div>\r\n The logarithmic function is defined only when the input is positive, so this function is defined when [latex]x+3>0[\/latex]. Solving this inequality,<\/p>\r\n\r\n The domain of [latex]f\\left(x\\right)={\\mathrm{log}}_{2}\\left(x+3\\right)[\/latex] is [latex]\\left(-3,\\infty \\right)[\/latex].<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n What is the domain of [latex]f\\left(x\\right)={\\mathrm{log}}_{5}\\left(x - 2\\right)+1[\/latex]?<\/p>\r\nSolution<\/a>\r\n\r\n<\/div>\r\n What is the domain of [latex]f\\left(x\\right)=\\mathrm{log}\\left(5 - 2x\\right)[\/latex]?<\/p>\r\n\r\n<\/div>\r\n The logarithmic function is defined only when the input is positive, so this function is defined when [latex]5 - 2x>0[\/latex]. Solving this inequality,<\/p>\r\n\r\n The domain of [latex]f\\left(x\\right)=\\mathrm{log}\\left(5 - 2x\\right)[\/latex] is [latex]\\left(-\\infty ,\\frac{5}{2}\\right)[\/latex].<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n What is the domain of [latex]f\\left(x\\right)=\\mathrm{log}\\left(x - 5\\right)+2[\/latex]?<\/p>\r\nSolution<\/a>\r\n\r\n<\/div>","rendered":" Before working with graphs, we will take a look at the domain (the set of input values) for which the logarithmic function is defined.<\/p>\n Recall that the exponential function is defined as [latex]y={b}^{x}[\/latex] for any real number x<\/em>\u00a0and constant [latex]b>0[\/latex], [latex]b\\ne 1[\/latex], where<\/p>\n In the last section we learned that the logarithmic function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] is the inverse of the exponential function [latex]y={b}^{x}[\/latex]. So, as inverse functions:<\/p>\n Transformations of the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] behave similarly to those of other functions. Just as with other parent functions, we can apply the four types of transformations\u2014shifts, stretches, compressions, and reflections\u2014to the parent function without loss of shape.<\/p>\n In Graphs of Exponential Functions<\/a> we saw that certain transformations can change the range<\/em> of [latex]y={b}^{x}[\/latex]. Similarly, applying transformations to the parent function [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] can change the domain<\/em>. When finding the domain of a logarithmic function, therefore, it is important to remember that the domain consists only of positive real numbers<\/em>. That is, the argument of the logarithmic function must be greater than zero.<\/p>\n For example, consider [latex]f\\left(x\\right)={\\mathrm{log}}_{4}\\left(2x - 3\\right)[\/latex]. This function is defined for any values of x<\/em>\u00a0such that the argument, in this case [latex]2x - 3[\/latex], is greater than zero. To find the domain, we set up an inequality and solve for\u00a0x<\/em>:<\/p>\n In interval notation, the domain of [latex]f\\left(x\\right)={\\mathrm{log}}_{4}\\left(2x - 3\\right)[\/latex] is [latex]\\left(1.5,\\infty \\right)[\/latex].<\/p>\n What is the domain of [latex]f\\left(x\\right)={\\mathrm{log}}_{2}\\left(x+3\\right)[\/latex]?<\/p>\n<\/div>\n The logarithmic function is defined only when the input is positive, so this function is defined when [latex]x+3>0[\/latex]. Solving this inequality,<\/p>\n The domain of [latex]f\\left(x\\right)={\\mathrm{log}}_{2}\\left(x+3\\right)[\/latex] is [latex]\\left(-3,\\infty \\right)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n What is the domain of [latex]f\\left(x\\right)={\\mathrm{log}}_{5}\\left(x - 2\\right)+1[\/latex]?<\/p>\n Solution<\/a><\/p>\n<\/div>\n What is the domain of [latex]f\\left(x\\right)=\\mathrm{log}\\left(5 - 2x\\right)[\/latex]?<\/p>\n<\/div>\n The logarithmic function is defined only when the input is positive, so this function is defined when [latex]5 - 2x>0[\/latex]. Solving this inequality,<\/p>\n The domain of [latex]f\\left(x\\right)=\\mathrm{log}\\left(5 - 2x\\right)[\/latex] is [latex]\\left(-\\infty ,\\frac{5}{2}\\right)[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n What is the domain of [latex]f\\left(x\\right)=\\mathrm{log}\\left(x - 5\\right)+2[\/latex]?<\/p>\n Solution<\/a><\/p>\n<\/div>\n\n\t\t\t \r\n\t
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How To: Given a logarithmic function, identify the domain.\r\n<\/strong><\/h3>\r\n
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Example 1: Identifying the Domain of a Logarithmic Shift<\/h3>\r\n
Solution<\/h3>\r\n
Try It 1<\/h3>\r\n
Example 2: Identifying the Domain of a Logarithmic Shift and Reflection<\/h3>\r\n
Solution<\/h3>\r\n
Try It 2<\/h3>\r\n
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\n
How To: Given a logarithmic function, identify the domain.
\n<\/strong><\/h3>\n\n
Example 1: Identifying the Domain of a Logarithmic Shift<\/h3>\n
Solution<\/h3>\n
Try It 1<\/h3>\n
Example 2: Identifying the Domain of a Logarithmic Shift and Reflection<\/h3>\n
Solution<\/h3>\n
Try It 2<\/h3>\n
Candela Citations<\/h3>\n\t\t\t\t\t