Question

The solution set of $\ln (5 - 7x) \leqslant 1$ is given by,
A. $\left[ {\dfrac{{5 - e}}{7},\dfrac{5}{7}} \right)$
B. $\left[ {\dfrac{{2 - e}}{3},\dfrac{2}{3}} \right)$
C. $( - 10,7)$
D. All real numbers

Answer 1

The set containing all the solutions of an equation is called the solution set for that equation. We can find the interval of $x$ by solving the inequality.
Use the property of logarithm;
$\Rightarrow$ ${e^{\ln (x)}} = x$
Logarithmic function and exponential function are the inverse function.

Complete step-by-step answer:
We are asked to find the solution set of $\ln (5 - 7x) \leqslant 1$.
Remove the logarithm by taking exponential both sides of the inequality $\ln (5 - 7x) \leqslant 1$.
$\Rightarrow$ $(5 - 7x) \leqslant e$
Logarithmic function and exponential function are the inverse function.
$\Rightarrow$ $5 - e \leqslant 7x$
Divide both the sides by $7$.
\Rightarrow$$$\dfrac{{5 - e}}{7} \leqslant \dfrac{{7x}}{7}$$ \Rightarrow$$$ \Rightarrow \dfrac{{5 - e}}{7} \leqslant x \ldots (1) All the solutions of $x$ greater than\dfrac{{5 - e}}{7}$$.
Logarithmic function is defined for positive values.
$\Rightarrow$ $5 - 7x > 0$
$\Rightarrow$ $5 > 7x$
$\Rightarrow$ $\dfrac{5}{7} > x \ldots (2)$
From the inequalities $(1)$ and $(2)$.
$\Rightarrow$ $\dfrac{{5 - e}}{7} < x < \dfrac{5}{7}$
The solution set of $\ln (5 - 7x) \leqslant 1$ is $\left[ {\dfrac{{5 - e}}{7},\dfrac{5}{7}} \right)$.

Correct Answer: $\left[ {\dfrac{{5 - e}}{7},\dfrac{5}{7}} \right)$

Note:
The logarithmic function is defined for positive values.
Use the property of logarithm;
$\Rightarrow$ ${e^{\ln (x)}} = x$