How To Add Ln Functions

to demonstrate the property of logarithms, we're asked to expand this logarithm as much as possible. we have natural log of x to the fourth, y to the second x the fifth root of x to the third, y to the fourth.

How To Add Ln Functions, but notice how we have x's outside the radical and underneath the radical, as well as y's outside the radical and underneath the radical.

so there's couple ways of doing this, but what i'm going to do is rewrite this radical using rational exponents. so we can write this as natural log, leave x to the fourth and y to the second alone for right now. but then the fifth root of x to the third is the same as x to the 3/5 power. and the fifth root of y to the fourth

is the same as y to the 4/5 power. now that it's in this form we can multiply x to the fourth and x to the 3/5, as well as y to the second and y to the 4/5. remember when multiplying when the bases are the same, we add the exponents. so to multiply x to the fourth and x to the 3/5, we need to add the exponents. well, 4 + 3/5 is just 4 and 3/5.

let's go ahead and convert this to an improper fraction. 5 x 4 is 20 + 3 that would be 23/5. so we'd have natural log of x to the 23/5. then we'll have y to the power of 2 + 4/5. well, 2 + 4/5 is just 2 and 4/5. convert this to an improper fraction, 5 x 2 + 4 that's 14/5. so we have y to the 14/5. now, notice how we have natural log of a product,

so we can expand this further using the product property of logarithms given here. so this is going to be = to natural log of x to the 23/5 + because we have a product natural log y to the 14/5. and now we can expand this one more time by using the power property of logarithms given here where we have log base b of x raised to the power of n is = to n x log base b of x. so we can take this exponent

and move it to the front of the logarithm here and here. so in expanded form we would have 23/5 natural log x + 14/5 natural log y. this would be the given logarithm expanded as much as possible. â