Question

prove that log xy base a























is equal to log x base A + log Y base a ​

Answer 1

Answer:

If a is the base of the logarithm, then raise both the sides separately to the exponent on a. Applying the rule of indices we get both sides as xy as by definition of logarithm of x to the base 'a', we have a^(log_a)(x) =x. Thereafter we can conclude the result from the injectivity of the power function.

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Answer 2

Answer:

Put

u = logₐ x ,    v = logₐ y    and     w = logₐ xy.

We need to show that w = u + v.

For this, notice

$a^w = xy = (a^u)(a^v)=a^{u+v}$

From here it follows that w = u + v, as required.

Hope this helps!