Question

Compute for the value of x in a given logarithmic inequalities log2(x+1)>log4(x2)​

Answer 1

Step-by-step explanation:

Let, income be 7x

and expenditure be 6x

So savings =7x−6x=x

7x=14000

x=2000

So savings =2000

Answer 2

Given:

$\log _{2}(x+1)>\log _{4}\left(x^{2}\right)$

Here, the bases are different, but they are related by the fact that 4=2²

$\log _{4}\left((x+1)^{2}\right)>\log _{4}\left(x^{2}\right)$

$\text { So, }(x+1)^{2}>x^{2}, \text { implying that } 2 x+1>0 \Longrightarrow x>-\frac{1}{2}$

the arguments of each logarithm must be positive, which excludes the case x=0.

Therefore, the final solution set is

$x>-\frac{1}{2}, x \neq 0$

Hence the answer is $x>-\frac{1}{2}, x \neq 0$