```Question 282953
(Please use "sqrt" for square roots and not the half-hearted radical: &#8730;. The later makes it nearly impossible to tell what the radicand is.)<br>
{{{log(a, (a/sqrt(x))) - log(a, (sqrt(ax)))}}}
Making this into a single logarithm is the easy part. This expression exactly matches the pattern of the left side of the following property of logarithms:
{{{log(a, (p)) - log(a, (q)) = log(a, (p/q))}}}
So we can use the right side of the property to replace your expression with a single logarithm:
{{{log(a, ((a/sqrt(x))/sqrt(ax)))}}}
Now we simplify the argument. We want to a) simplify the fraction within a fraction to a "regular" fraction; b) make sure denominators are rational; and do any other kinds of simplifying which can be done. There are several paths to these goals. But there is one which is probably the quickest. If we multiply the numerator and denominator of the "big" fraction by its denominator a lot of good things happen all at once, as you'll see:
{{{log(a, (((a/sqrt(x))/sqrt(ax))(sqrt(ax)/sqrt(ax)))))}}}
which simplifies as follows:
{{{log(a, ((a*sqrt(ax)/sqrt(x))/(sqrt(ax))^2))}}}
{{{log(a, ((a*sqrt(a)*sqrt(x)/sqrt(x))/ax))}}}
{{{log(a, ((a*sqrt(a)*cross(sqrt(x))/cross(sqrt(x)))/ax))}}}
{{{log(a, (a*sqrt(a)/ax))}}}
We've gotten rid of the fraction within a fraction and the irrational denominators all at once! But there is more we can simplify:
{{{log(a, (cross(a)*sqrt(a)/cross(a)x))}}}
{{{log(a, (sqrt(a)/x))}}}
This is probably the desired answer. But, because the base of the logarithm is "a" and because of the square root of "a" is a power of "a", we can reduce this further:
{{{log(a, (a^(1/2)/x))}}}
Using the earlier property (this time to split these logarithms apart):
{{{log(a, (a^(1/2))) - log(a, (x))}}}
Now we can use another property, {{{log(a, (p^q)) = q*log(a, (p))}}}, we can move the exponent on a out in front:
{{{(1/2)log(a, (a)) - log(a, (x))}}}
And since {{{log(a, (a)) = 1}}} by definition this becomes:
{{{1/2 - log(a, (x))}}}
This is a simpler expression than {{{log(a, (sqrt(a)/x))}}} and it does include just a single logarithm. But it does not express the original expression "as a single logarithm" so it may not be considered "correct" given the instructions. ```