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### Section 6.2 : Logarithm Functions

22. Sketch the graph of $$g\left( x \right) = \ln \left( {x + 5} \right)$$ .

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For this problem all we need to do is recall the Transformations section from a couple of chapters ago. Using the “base” function of $$f\left( x \right) = \ln \left( x \right)$$ the function for this part can be written as,

$g\left( x \right) = \ln \left( {x + 5} \right) = f\left( {x + 5} \right)$

Therefore, the graph for this part is just the graph of $$f\left( x \right)$$ shifted left by 5.

The graph of this function is shown below. The blue dashed line is the “base” function, $$f\left( x \right)$$, and the red solid line is the graph for this part, $$g\left( x \right)$$. Do not get excited about the fact that we plugged negative values of $$x$$ into the function! The problem with negative values is not the values we plug into a logarithm. Instead the problem with negative values is when we go to evaluate the logarithm.

It is perfectly fine to plug negative values into a logarithm as long as we don’t end up actually evaluating a negative number. So, in this case we can see that as long as we require $$x > - 5$$ then $$x + 5 > 0$$ and so those are acceptable values of $$x$$ to plug in since we aren’t going to evaluate negative number in the logarithm.

Note however that we do have avoid $$x < - 5$$ since that would mean evaluating logarithms at negative numbers.