Question 1153723
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We can have function(s) of the form<br>
{{{log(5,(x+b))+c}}}<br>
Given that the function is to have an integer value at x=-3, we want -3+b to be a power of 5.<br>
(a) first example
5^0 = 1; -3+b = 1  -->  b = 4<br>
{{{log(5,(-3+4)) = log(5,1) = 0}}}<br>
We want the function value at -3 to be -2, so we need to subtract 2.  Our first function that satisfies the requirements is<br>
{{{log(5,(x+4))-2}}}<br>
A graph (window -10 to 10; -5 to 2):<br>
{{{graph(400,400,-10,10,-5,2,log(5,(x+4))-2)}}}<br>
(b) second example
5^1 = 5; -3+b = 5  -->  b = 8<br>
{{{log(5,(-3+8)) = log(5,5) = 1}}}<br>
We want the function value at -3 to be -2, so we need to subtract 3.  Our second function that satisfies the requirements is<br>
{{{log(5,(x+8))-3}}}<br>
A graph (window -10 to 10; -5 to 2), showing both graphs passing through (-3,-2):<br>
{{{graph(400,400,-10,10,-5,2,log(5,(x+4))-2,log(5,(x+8))-3)}}}<br>
We can also have functions of the form<br>
{{{a*log(5,(x+b))}}}<br>
Again using b=8,<br>
{{{log(5,(-3+8)) = log(5,5) = 1}}}<br>
We want the function value to be -2 at x=-3; so we can multiply this function by -2.  Our third function that satisfies the requirements is<br>
{{{(-2)*log(5,(x+8))}}}<br>
A graph again, showing the graphs of all three functions passing through (-3,-2):<br>
{{{graph(400,400,-10,10,-5,2,log(5,(x+4))-2,log(5,(x+8))-3,(-2)*log(5,(x+8)))}}}<br>
You should be able to see that we can get as many functions as we want that satisfy the requirement.<br>
Here is one more that gives the value -2 at x=-3:<br>
{{{(-5)*log(5,(x+8))+3}}}<br>
One more graph, showing all four functions passing through (-3,-2):<br>
{{{graph(400,400,-10,10,-5,2,log(5,(x+4))-2,log(5,(x+8))-3,(-2)*log(5,(x+8)),(-5)*log(5,(x+8))+3)}}}<br>