Question

How do you solve $\ln (x - 6) - \ln (5) = \ln (7) + \ln (x - 2)$ ?

Answer 1

Hint : In order to simplify the natural log equation with the formula is $\ln (x \cdot z) = \ln x + \ln z$ and $\ln \dfrac{x}{z} = \ln x - \ln z$ .The natural logarithm of a number is its logarithm to the base of the mathematical constant. The natural logarithm of $x$ is generally written as $\ln x,{\log _e}x$ or sometimes, if the base $e$ is implicit, simply $\log x$ . We get the required solution by comparing the formula.
Formula:
$\ln (x \cdot z) = \ln x + \ln z$
$\ln \dfrac{x}{z} = \ln x - \ln z$ .

Complete step-by-step answer :
In this problem,
The natural logarithm of the equation is $\ln (x - 6) - \ln (5) = \ln (7) + \ln (x - 2)$ . First, we compare the equation with the formula is $\ln (x \cdot z) = \ln x + \ln z$ and $\ln \dfrac{x}{z} = \ln x - \ln z$ .
LHS $=$ RHS
$\ln (x - 6) - \ln (5) = \ln (7) + \ln (x - 2)$
Comparing LHS equation, $\ln (x - 6) - \ln (5)$ with the formula $\ln \dfrac{x}{z} = \ln x - \ln z$ , Subtraction of the log is the result of the source values being divided, where $x = (x - 6),z = 5$
LHS: $\ln (x - 6) - \ln (5) = \ln \left( {\dfrac{{x - 6}}{5}} \right)$
and Comparing RHS equation, $\ln (7) + \ln (x - 2)$ with the formula $\ln (x \cdot z) = \ln x + \ln z$ ,Addition of logs is the consequence of the source values being multiplied. So
RHS: $\ln (7) + \ln (x - 2) = \ln (7(x - 2))$ .
Combining all this equation together, we can get
$\ln \left( {\dfrac{{x - 6}}{5}} \right) = \ln (7(x - 2)$
By remove the inverse of log on both side, we get
$\left( {\dfrac{{x - 6}}{5}} \right) = (7(x - 2)$
Expanding the bracket on RHS and multiply by $5$ on both sides, we have

x - 6 = 35(x - 2) \\\ x - 6 = 35x - 70 \; \ $$ By simplify the equation to get the value of $$x$$ , we get $$x - 35x = 6 - 70 \Rightarrow - 34x = - 54$$ $$x = \dfrac{{54}}{{34}} = \dfrac{{27}}{{17}} = 1.588$$ Therefore the required solution, $$x = 1.588$$ . **So, the correct answer is “$$x = 1.588$$ ”.** **Note** : First we have to plot a graph with respect to the problem is shown below. The graph of the red curve represent the equation $$\ln (x - 6) - \ln (5)$$ The graph of the blue curve represent the equation $$\ln (x - 6) - \ln (5)$$ The graph of the green line is $$x = 1.588$$  In this problem, we have to remember the natural logarithm formula $$\ln (x \cdot z) = \ln x + \ln z$$ and $$\ln \dfrac{x}{z} = \ln x - \ln z$$ on comparing with the given equation to get the required value $$x$$ .