Let u = ln a and v = ln b. Write each expression in terms of u an...

Question

Let u = ln a and v = ln b. Write each expression in terms of u and v without using the ln function. ln √(a^3/b^5)

Accepted Answer

Hello, everyone. We are asked to write the given logarithmic expression in terms of P and Q, where P equals the natural log of X and Q equals the natural log of Y. The expression we are given is the natural log of in parentheses. X raised to the fourth power divided by Y raised to the seventh power. Our answer choices are a seven P plus four, Q B seven P minus four, Q C four, P minus seven Q D four P plus seven, Q, I'm gonna rewrite my expression and the division in it as a fraction. So I have the natural log of my numerator which is x-rays to the fourth power. And then the denominator is why raised to the seventh power. I recall that when I think of the rules of logarithms that I have a rule that says that if I'm taking the log of a division problem, I can rewrite this as two separate logarithms subtracted. So what this would look like is this is the natural log of X raised to the fourth power minus the natural log of Y raised to the seventh power. Next recall that when we're working with logarithms, our exponents can be brought to the front as a coefficient. So rewriting this, I now have four multiplied by the natural log of X minus seven, multiplied by the natural log of Y. At this step is where I can substitute in P and Q. So I will put P in for the natural log of X. So I now have four P minus and I will put Q in for the natural log of Y. So this is now minus seven Q. So this expression gets rewritten as four P minus seven Q. That is our answer and that's answer choice C have a nice day.

Let u = ln a and v = ln b. Write each expression in terms of u and v without using the ln function. ln √(a^3/b^5)

Key Concepts

Properties of Logarithms

Change of Base Formula

Exponents and Roots

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