SOLUTION: Write as a single logarithm: (1/2) log7 W - (2/3) log7 X + (3/4) log7 Y

SOLUTION: Write as a single logarithm: (1/2) log7 W - (2/3) log7 X + (3/4) log7 Y - (4/5) log7 Z

Question 814155: Write as a single logarithm:
(1/2) log7 W - (2/3) log7 X + (3/4) log7 Y - (4/5) log7 Z
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
The following properties of logarithms are often used in problems like this:

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.

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:

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:

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:

The first two logs of what remains have a "+" between them so we use the first property:

The remaining two logs have a "-" between them so back to the second property:

which simplifies to:

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