Positive Values of $a$, $b$, $c$, and $d$ for Log Inequality

[anemone]

Determine the positive values of $a,\,b,\,c$ and $d$ such that $\log_{d+3} (a + 3)>\log_{c+2} (d + 2)>\log_{b+1} (c + 1) >\log_a (b)$.

 

[jvicens]

To solve this problem, we can use the properties of logarithms to rewrite the inequalities in a more manageable way. First, let's rewrite the first inequality as a logarithm equation:

$\log_{d+3} (a + 3) > \log_{c+2} (d + 2)$

Using the property $\log_a (b) > \log_a (c)$ if $b > c$, we can rewrite this as:

$a + 3 > d + 2$

Similarly, we can rewrite the second inequality as:

$d + 2 > c + 1$

And the third inequality as:

$c + 1 > b$

Now we have a system of three inequalities:

$a + 3 > d + 2$

$d + 2 > c + 1$

$c + 1 > b$

To solve this system, we can start by setting $b = 1$ since we know that the logarithm of 1 is always 0. This simplifies the third inequality to:

$c > 0$

Next, we can set $c = 2$ to simplify the second inequality to:

$d > 1$

Finally, we can set $d = 3$ to simplify the first inequality to:

$a > 2$

Therefore, the values of $a,\,b,\,c,$ and $d$ that satisfy the given inequalities are:

$a > 2,\, b = 1,\, c > 0,\, d > 1$

In other words, any positive values for $a,\,c,$ and $d$ that are greater than 2, 0, and 1 respectively, and any value for $b$ that is equal to 1, will satisfy the given inequalities.