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 (b^4√a)

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 the fifth power multiplied by the cube root of Y. Our answer choices are a five P minus one third, Q B one third, P minus five Q C, one third, P plus five Q D five P plus one third Q before we substitute in any PS and Qs, I want to simplify my logarithmic expression. So we have again, the natural log of in parentheses X raised to the fifth power multiplied by the third root of Y. I recall that when working with logarithms, I can break multiplication up into the summ of two logs, meaning instead of having things multiplied together, I can rewrite this as the natural log of X raised the fifth power plus the natural log of the cube root of Y. Next, I'm gonna rewrite my radical expression, the cube root of Y in terms of an exponent. So I will still have the natural log of X rays to the fifth power plus. Now the natural log and recall that the cube root is the same as an exponent of one third. So we have the natural log of Y raised to the one third power. Next, recall the logarithm rule that says that if the value has an exponent, we can bring it out front as a coefficient. That's what I'm gonna do in both cases. So this can be rewritten as five natural log of X plus one third, natural log of Y. At this point, I am going to substitute in the P and the Q recall P was the natural log of X. So I can rewrite this now as five P plus Q was the natural log of Y. So the one third and now that's gonna be multiplied by Q because we're subbing it in for the natural log of Y. So our answer here in terms of P and Q is five P plus one third, Q and that is answer choice D have a nice day.

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Let u = ln a and v = ln b. Write each expression in terms of u and v without using the ln function. ln (b^4√a)

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