Question

If x,y ∈ 2 n when n ∈ I and 1+log e ​ y=log 2 ​ y, then the value of (x+y) is

Answer 1

The value of (x + y) is 2n+1.

Explanation:

To find the value of (x + y), let's solve the given equation and substitute the obtained values into the expression.

1 + logₑ​(y) = log₂​(y)

We can rewrite the equation using logarithmic properties:

logₑ​(e) + logₑ​(y) = log₂​(y)

Using the fact that logₐ​(a) = 1, we simplify further:

1 + logₑ​(y) = log₂​(y)

Or, logₑ​(y) + 1 = log₂​(y)

Now, let's convert the equation to the same base logarithm:

logₑ​(y) + logₑ​(e) = log₂​(y)

Applying the logarithmic property logₐ​(b) = logₐ​(c) is equivalent to b = c, we have:

y⋅e = 2⋅y

Dividing both sides by y:

e = 2

Now we can substitute this value into the expression (x + y):

(x + y) = (2n + 2n) = 2n+1

Therefore, the value of (x + y) is 2n+1.

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