Question

if f(x) = log X then prove that f(mn)=f(m)+f(n)​

Answer 1

Given:

f(x) = log X

To Prove:

f (mn) = f (m) + f (n)

Solution:

→ The basic properties of the logarithm function are:

  (i) log (ab) = log(a) + log(b)

  (ii) log (a/b) = log(a) - log(b)

  (iii) log (a)ᵇ = b log(a)

→ In the given question:

 The given function is f(x) = log X

     ∴ f(mn) = log (mn)

     ∴ f(mn) = log (m) + log (n)

     ∴ f(mn) = f(m) + f(n)

Therefore if f(x) = log X then f (mn) = f (m) + f (n) Proved.

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Answer 2

Final answer:

The equality for the functions can be proved as  $f(mn)=f(m)+f(n)$ under the condition that $f(x)=log(x)$.

Given:

The mathematical statement in the form of $f(x)=log(x)$

To Find:

We are required to prove that as  $f(mn)=f(m)+f(n)$ for the above given condition that $f(x)=log(x)$

Explanation:

The logarithm is such an operator which when wisely applied on any mathematical expression or numeric value, changes the complexity of the mathematic problems from the most difficult level to the relatively easier form by rewriting them in the terms of exponential functions.

Note the following basic rule of the logarithm function which will be employed to solve this problem.

$log (ab) = log(a) + log(b)$

To prove the statement, start solving it from the left hand side (LHS)  and gradually show it equal to its right hand side (RHS) .

Step 1 of 3

Deduce the following using the given condition.

$f(m)=log(m)\\f(n)=log(n)$

Start from the left hand side (LHS) of the identity that is to be proved. Write the following.

$LHS=f(mn)$

Step 2 of 3

Now, put the value of $mn$ in place of $x$ in the given condition to get the following.

$LHS\\=f(mn)\\=log(mn)$

Step 3 of 3

In the final step, use the logarithm function referred above and write the following.

$LHS\\=f(mn)\\=log(mn)\\ = log(m) + log(n)$

With the reference to the first two equations of step 1, we get the following.

$LHS\\=f(mn)\\=log(mn)\\ = log(m) + log(n)\\=f(m)+f(n)\\=RHS$

Hence, it is proved that  $f(mn)=f(m)+f(n)$ for the condition $f(x)=log(x)$.

Know more from the following links.

https://brainly.in/question/1164782

https://brainly.in/question/15569865

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