Use the properties of logarithms to expand the following expression.log(z3)/√x5y)Each logarithm should involve only one variable and should not have any radicals or exponents. You may assume that all variables are positive.

3 log (z) -  1/2log (x) - 1/2 log (5) -1/2 log y

Step-by-step explanation:

log(z3)/√x5y)

= log (z3)   - log (√x5y)   [division being subtracted]

= 3 log (z) - {log (√x) + log (√5y)}    [ multiplication being added in the denominator]

= 3 log (z) - {log (x)^1/2 + log (5)^1/2+ log (y)^1/2}  

= 3 log (z) -  1/2{log (x) + log (5)+ log y }

 We can solve inside the curly bracket and then multiply with 1/2 or we can solve it separately

= 3 log (z) -  1/2log (x) - 1/2 log (5) -1/2 log y

How does this work?

1) any two terms being multiplied in logarithms are added with the log of each.

2) the exponent becomes the co- efficient.

3)  any two terms being divided in logarithms are subtracted with the log of each

The side x is the hypotenuse of the triangle.

And we know that the leg that is labeled with a line is also 6.

Use the Pythagorean theorem and solve for x.

a² + b² = x²
3² + 6² = x²
9 + 36 = x²
45 = x²
√45 = x
√9×5 = x
√9√5 = x
3√5 = x

The answer is 3√5

Hope this helps :)