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What is ln multiplied by ln?

Question:

What is {eq}\ln {/eq} multiplied by {eq}\ln {/eq}?

Algebraic Multiplication of 2 Variables:

Suppose that we are given two functions {eq}f(x) {/eq} and {eq}g(x) {/eq}:

  • In a general case, if {eq}f(x)\neq g(x) {/eq}, then the product cannot be simplified and we end up with {eq}f(x)g(x) = f(x)g(x) {/eq}
  • On the other hand, if {eq}g(x) = f(x) {/eq}, then we can simplify our product by {eq}f(x)g(x) = f(x)f(x) = f^2(x) {/eq}

Answer and Explanation:

Since the argument of the natural logarithm is not provided, let's consider two cases:

a. Suppose that we are multiplying logarithms involving different functions:

{eq}\ln (f(x))\cdot \ln(g(x)) {/eq}

In this specific case, there is no property that allows us to simplify the result, that is, this is the most simplified form we can obtain.

b. Suppose that the two arguments are same:

{eq}\ln (f(x))\cdot \ln (f(x)) = \ln^2 (f(x)) {/eq}

In this case, we use a simple property that:

{eq}a\cdot a = a^2 {/eq}

To conclude, there is no specific logarithmic property that would simplify the result. We rely on a simple algebraic square when we multiply two logarithms having same argument, that is:

{eq}\boxed{\ln(f(x))\cdot \ln(f(x)) = \ln^2 (f(x))} {/eq}


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