You can put this solution on YOUR website! (I am not sure what x^+3 and x^-1 mean. I am going to assume that they mean and . In the future use exponents around exponents which are more than 1 character long.)
Multiple logarithms can be combined into one using the following properties of logarithms:
All that is required is that the logarithms be the same base and that the coefficients are 1's. There is another property of logarithms that can used to change a coefficient which is not 1 into a 1:
1)
Both logarithms have coefficients that are not 1's. So we will use the third property to move the coefficient into the argument as an exponent:
Now the coefficients are 1's. Since this is an addition of logarithms and the bases of the logarithms are the same, we can use the first property above to combine them into one:
This may be an acceptable answer. But with some clever Algebra we can simplify the argument a little more. Since 1/2 = 2/4 we can write:
Since and we can rewrite this as:
Since :
Now we can combine the two fourth roots into one:
which simplifies to:
And finally we can simplify the combined fourth root by factoring out :
Normally we would use replace with . However, x had to be positive in the original equation (because the argument of must be positive), we do not need to use the absolute value:
This is a fully simplified answer.
2)
The second logarithm does not have a coefficient of 1 so we'll use the thrid property to move it to the argument as an exponent:
Since the bases of the logarithms are the same and this is a subtraction, we will use the second property above to combine them:
Since this can be rewritten as:
(