Question 814155
The following properties of logarithms are often used in problems like this:<ul><li>{{{log(a, (p)) + log(a, (q)) = log(a, (p*q))}}}</li><li>{{{log(a, (p)) - log(a, (q)) = log(a, (p/q))}}}</li><li>{{{n*log(a, (p)) = log(a, (p^n))}}}</li></ul>The first two will combine logs into a single log, the first for when there is a "+" between the logs and the second for when there is a "-" between. These two properties require that the bases of the logs be the same and that the coefficients of the logs are 1's. The third property can be used to "move" a coefficient that is not 1 into the argument as its exponent.<br>
{{{(1/2) log(7, (w)) - (2/3) log(7, (x)) + (3/4) log(7, (y)) - (4/5) log(7, (z))}}}
Since none of the logs in your expression have a coefficient of 1, we will start by using the third property to move those coefficients into the argument as the exponent:
{{{log(7, (w^(1/2))) - log(7, (x^(2/3))) +  log(7, (y^(3/4))) - log(7, (z^(4/5)))}}}
All those fractional exponents represent roots of various kinds. Since radicals display better on algebra.com I am going to rewrite all those roots in radical form before proceeding:
{{{log(7, (sqrt(w))) - log(7, (root(3, x^2))) +  log(7, (root(4, y^3))) - log(7, (root(5, z^4)))}}}<br>
Now we can start using the other two properties to combine these logs. The first two have a "-" between them so we use the second property:
{{{log(7, (sqrt(w)/root(3, x^2))) +  log(7, (root(4, y^3))) - log(7, (root(5, z^4)))}}}
The first two logs of what remains have a "+" between them so we use the first property:
{{{log(7, ((sqrt(w)/root(3, x^2))*root(4, y^3))) - log(7, (root(5, z^4)))}}}
The remaining two logs have a "-" between them so back to the second property:
{{{log(7, (((sqrt(w)/root(3, x^2))*root(4, y^3))/root(5, z^4)))}}}
which simplifies to:
{{{log(7, (sqrt(w)*root(4, y^3)/(root(3, x^2)*root(5, z^4))))}}}