find the domain of root of a logarithmic functionI'm a little confused about this question since output of a logarithmic function varies from − ∞ to ∞ .I should find the domain of this function: y = log x ⁡ ( 10 − x 2 ) . How can I find the interval that makes log x ⁡ ( 10 − x 2 ) greater than zero?

Question

find the domain of root of a logarithmic functionI&#039;m a little confused about this question since output of a logarithmic function varies from   −  ∞ to   ∞ .I should find the domain of this function:   y  =            log      x        ⁡    (    10    −          x      2        )   . How can I find the interval that makes       log    x    ⁡  (  10  −      x    2    ) greater than zero?

verzaadtwr · Accepted Answer

Recall for   a  &amp;gt;  0  ,  a  ≠  1  ,  b  &amp;gt;  0      log    a    ⁡  b  =            ln      ⁡      b              ln      ⁡      a      Thus we have  f  (  x  )  =      log    x    ⁡  (  10  −      x    2    )  =            ln      ⁡      (      10      −              x        2            )              ln      ⁡      x      Then   f  (  x  )  ≥  0 if and only if   10  −      x    2    ≥  1  ,  x  &amp;gt;  1 or   0  &amp;lt;  10  −      x    2    ≤  1  ,  0  &amp;lt;  x  &amp;lt;  1. Since the later case cannot happen, then we must have   10  −      x    2    ≥  1 and   x  &amp;gt;  1, which gives   1  &amp;lt;  x  ≤  3

Joel French · Accepted Answer

First of all, I would suggest to write your   log in a fixed basis      log    x    ⁡  (  10  −      x    2    )  =            ln      ⁡      (      10      −              x        2            )              ln      ⁡      (      x      )        .This changes your question to            ln      ⁡      (      10      −              x        2            )              ln      ⁡      (      x      )        &amp;gt;  0or equivalently:  ln  ⁡  (  x  )  ln  ⁡  (  10  −      x    2    )  &amp;gt;  0.Now, you have to find all roots of   10  −      x    2   which gives you the intervals where   ln  ⁡  (  10  −      x    2    )  &amp;gt;  0 and then you are almost finished.

find the domain of root of a logarithmic function I'm a little confused about this question since o

Answered question

Answer & Explanation

New Questions in Algebra II