Question

How do you solve $\ln \left( -m \right)=\ln \left( m+10 \right)$ ?

Answer 1

Hint: The given problem can be solve by using the basic property of logarithms, which is
$\ln a-\ln b=\ln \left( \dfrac{a}{b} \right)$
At first we bring the two terms on one side and then implement the property to make a single logarithmic term. Finally, we take exponents on both sides and evaluate to get the final answer.

Complete step by step answer:
The given equation in $m$ is
$\ln \left( -m \right)=\ln \left( m+10 \right)$
We start the solution by bringing the $\ln \left( m+10 \right)$ term on the LHS.
$\Rightarrow \ln \left( -m \right)-\ln \left( m+10 \right)=0....equation1$
Now, we need to apply the property of logarithms which states that
$\ln a-\ln b=\ln \left( \dfrac{a}{b} \right)....equation2$
Comparing $equation1$ with $equation2$ , we get
$a=m$ and $b=m+10$
Therefore, $equation1$ can be simplified as
$\Rightarrow \ln \left( \dfrac{-m}{m+10} \right)=0$
Now, we take exponents on both sides of the equation. Then,
$\Rightarrow {{e}^{\ln \left( \dfrac{-m}{m+10} \right)}}={{e}^{0}}....equation3$
We all know the property of exponent and logarithms which states that,
${{e}^{\ln x}}=x....equation4$
Comparing $equation3$ with $equation4$ , we get,
$x=\dfrac{-m}{m+10}$
Therefore, simplifying $equation3$ accordingly, we get,
$\Rightarrow \dfrac{-m}{m+10}={{e}^{0}}$
${{e}^{0}}$ is nothing but $1$ . Then,
$\Rightarrow \dfrac{-m}{m+10}=1$
Multiplying $\left( m+10 \right)$ on both sides of the equation, we get,
$\Rightarrow -m=m+10$
Adding $m$ on both sides of the equation, we get,
$\Rightarrow 0=2m+10$
Subtracting $10$ from both sides of the equation, we get,
$\Rightarrow -10=2m$
Dividing both sides of the equation by $2$ , we get
$\Rightarrow -5=m$

Therefore, we can conclude that the solution of the given equation is $m=-5$.

Note: Students must be careful while dealing with the addition and subtraction of logarithmic terms as they often multiply the entire terms and end up with wrong answers. Another way to solve this problem is by taking the exponent on both sides in the first step and then equate the powers of the exponent on both sides.