Question

Solve ${\log _x}(x + 7) < 0$.

Answer 1

Hint: The given question is solved based on the logarithm function. In that function a particular condition is applied ${\log _x}(x + 7) < 0$ i.e. the given function is less than zero.

Complete step by step answer:
Logarithm function is the inverse function to exponentiation, which means the logarithm of a given number $x$ is the fixed number. If $a$ the base then it must be raised to produce that number $x$.
The logarithm function can be written as $\log $ where $a$ and $b$ are the variables and $1,2,3,$ …….. so on called the constants.
The graph of a log function includes the set of all posture real numbers, when no base is written.
We have ${\log _x}(x + 7) < 0$ is a given function. In which logarithm is applied and $x$ is variable, $y$ is a constant.
In the given equation condition is applied to the function i.e. it is less than zero.
Consider that $x$ lies between $0$ and $1$ i.e. $0 < x < 1$ which means $x$ is less than $1$ and greater than zero.
For $0 < x < 1$
We have given that ${\log _x}(x + 7) < 0$
We can say that the function $x + 7$ is greater than $7$.
let $x + 7 > 1$ ---------(A)
Now solving the above term by separating it .
As $x > 0$, so, we can subtract $1$ on both sides of equation $(A)$ we get.
$x + 7 - 1 > 1 - 1$
$x + 6 > 0$
Now if $x > 0$ then $x + 6 > 0$.
But the value of $x$ can be find by solving the above term
$x > - 6$

So we conclude the result that $x$ lies between $0$ and $1$.
So, $x \in (0,1)$ hence proved the result.

Note: In the given function, is based on the logarithm function. There are two main reasons to use logarithmic scales in charts and graphs. If the equation $y - \log b(x)$ means that $y$ is passed or exponent that $b$ is raised to in order to get $x$. Logarithmic function $\log (x)$ grows very slowly for large $x$. Logarithmic scales are used to compress large – scale scientific data.