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Thread: Simple Question About "Graphs of Logarithmic Function"

  1. #1
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    Question Simple Question About "Graphs of Logarithmic Function"

    First of all, glad to be one of your members.

    Simple Question About "Graphs of Logarithmic Function"-untitled.jpg

    Can you please explain to me the first row and the second one! I didn't get it yet.

    This is the graph:

    Simple Question About "Graphs of Logarithmic Function"-untitled2.jpg
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  2. #2
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    domain means what x values are rellevant for this function

    range means what y values are possible as a result of the function

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  3. #3
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    In this case the domain of $\displaystyle f(x)= e^x$ is all real numbers because we can calculate $\displaystyle 2^{x}$ for x any real number. And, of course, the result of that calculation is always a positive number and can be any positive real number. That is why the range is all positive real numbers.

    For $\displaystyle f(x)= log_2(x)$, the domain is all positive numbers because $\displaystyle log_2(x)$ (or logarithm of any base) is defined for all positive numbers but not for 0 or any negative numbers. If $\displaystyle 0< x< 1$, though, $\displaystyle log_2(x)< 0$ and $\displaystyle log_2(1)= 0$ so the range is all real numbers.

    In fact, for any positive number, a, $\displaystyle a^x$ and $\displaystyle log_a(x)$ are inverse functions: if $\displaystyle y= a^x$ then $\displaystyle x= log_a(y)$ so they "swap" domain and range. That is also why the graphs are symmetric about the line y= x.
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