SOLUTION: F(x)=8+6*log base 9 (x-7) find the domain and range

Question 380156: F(x)=8+6*log base 9 (x-7)
find the domain and range
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!

When determining a domain, the main reasons (IOW this list is not complete) certain numbers are excluded from the domain are:

Zeros in denominators
Negative radicands (expressions within a radical) of even-numbered roots (square roots, 4th roots, 6th roots, etc.)
Zero or negative bases or arguments of logarithms

Your function has no denominators or even-numbered roots. But it does have a logarithm. So we must make sure that all numbers in the domain make both the base and the argument of the logarithm positive. Your base is 9 so there is no problem there. But your argument, (x-7), contains the variable so we must make sure it stays positive:
x-7 > 0
Solving this we get:
x > 7
This is our domain: All real numbers that are greater than 7.

To find the range, one usually needs to determine what possible values the function (or y) will have when x takes on all the different values of the domain.

Even though the base and argument of a logarithm must be positive, the logarithm itself can be any real number. And if the logarithm in your function can be any real number then your function's value (which is 6 times a logarithm plus 8) will also end up being any real number. So the range of your function is all real numbers.

Here's a graph of your function to illustrate all this:

Note how the graph is completely to the right of x=7. IOW, all points on the graph have an x-coordinate that is greater than 7. On the other hand the y values range from negative infinity (near x = 7) to positive infinity (for large x values) and everything in between (i.e. there are no "gaps" as it goes from negative to positive infinity).

If it appears that the graph include points where x=7 this is only due to an imperfection's in Algebra.com's graphing software and/or your video hardware and software. It may also appear that the graph does not go upwards or downwards forever. Again, the previously mentioned imperfections are to blame. The graph does keep going up and down forever.

If you have a graphing calculator and you want to try to graph this function, then use:
y = 8 + 6*log(x-7)/log(9)
which is the same function as yours except that it use an expression of base 10 logarithms (which you calculator "knows") which is equivalent to the base 9 logarithm.
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