# How do you find the domain and range of log_(6) (49-x^2)?

May 30, 2017

The argument of a $\log$ function must be positive.

#### Explanation:

So
$49 - {x}^{2} > 0 \to {x}^{2} < 49$

This only happens if
$- 7 < x < + 7$ this is the domain

Range:
With $x$ nearing $\pm 7$ the argument $49 - {x}^{2}$ will be nearing $0$ and the $\log$ itself will go to $- \infty$

Or, in the language:
${\lim}_{x \to \pm 7} {\log}_{6} \left(49 - {x}^{2}\right) = - \infty$

The top of the range is when the argument is maximal, this means when $x = 0$, the max value will be:
${\log}_{6} 49 = {\log}_{10} \frac{49}{\log} _ 10 6 \approx 2.172$
graph{log(49-x^2)/log(6) [-12.33, 12.99, -7.11, 5.55]}