Question

if log (a+b/2) =1/2(loga+logb) prove a=b

Answer 1

$log( \frac{a + b}{2} ) = \frac{1}{2} ( log(a) + log(b) ) \\ log( \frac{a + b}{2} ) = log( \sqrt{ab} ) \\ \frac{a + b}{2} = \sqrt{ab} \\ { (\frac{a + b}{2} )}^{2} = ab \\ {(a + b)}^{2} = 4ab \\ {a}^{2} + 2ab + {b}^{2} = 4ab \\ {a}^{2} - 2ab + {b}^{2} = 0 \\ {(a - b)}^{2} = 0 \\ a - b = 0 \\ a = b$

Answer 2

Prove that a=b, its true.

What is a logarithm in simple terms?

• A logarithm is the power to which a number must be raised in order to get some other number.

According to the question:

$\begin{aligned}&\log \left(\frac{a+b}{2}\right)=\frac{1}{2}(\log (a)+\log (b)) \\&\log \left(\frac{a+b}{2}\right)=\log (\sqrt{a b}) \\&\frac{a+b}{2}=\sqrt{a b} \\&\left(\frac{a+b}{2}\right)^{2}=a b \\&(a+b)^{2}=4 a b \\&a^{2}+2 a b+b^{2}=4 a b \\&a^{2}-2 a b+b^{2}=0 \\&(a-b)^{2}=0 \\&a-b=0 \\&a=b\end{aligned}$

Hence, Prove that a=b, its true.

Learn more about logarithm here,

https://brainly.in/question/3137322?msp_poc_exp=5

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