Question

How do you solve $\ln x + \ln 5 = \ln 10$ ?

Answer 1

Hint: This question can be solved by using the formula of logarithms. The formula of logarithm which we will use to solve this question: $\ln a - \ln b = \ln \left( {\dfrac{a}{b}} \right)$. Then, take the exponential on the equation to get the value of $x$.

Complete Step by Step Solution:
From the question, we have been given that we have to find the value of $x$ in $\ln x + \ln 5 = \ln 10$.
So, we have to use the above basic formula which we already discussed above to find the value of $x$ in the equation given in the question.
Therefore, the formula which we have to use and was discussed above is –
$\ln a - \ln b = \ln \left( {\dfrac{a}{b}} \right)$
According to the question, the given equation of logarithm is –
$\ln x + \ln 5 = \ln 10 \cdots \left( 1 \right)$
Now, we will use the transposition method to shift $\ln 5$ to another side with $\ln 10$.
As we already know that, in this method the function of operation becomes the opposite. Therefore, $\ln 5$ when shifts to another side will change into $ - \ln 5$. Hence, using the transposition method in equation (1), we get –
$ \Rightarrow \ln x = \ln 10 - \ln 5$
Now, using the formula of the logarithm, we get –
$ \Rightarrow \ln x = \ln \left( {\dfrac{{10}}{5}} \right)$
Further solving, we get –
$ \Rightarrow \ln x = \ln 2$
Now, taking exponential on both sides of the above equation, we get –
$ \Rightarrow {e^{\ln x}} = {e^{\ln 2}}$
Since we know that logarithm and exponential both are inverses of each other. So, if we have, for example, ${e^{\ln z}}$ then, it can also be written as $z$.
Therefore, the above equation can be written as –
$ \Rightarrow x = 2$

Hence, the value of $x$ in the equation given in the question is 2.

Note:
We all should be very well aware with the logarithms and formula of logarithms to solve this question. There are many other formulas of logarithms which can be used to solve other types of questions of logarithms, they are given as $\ln a + \ln b = \ln \left( {a \times b} \right)$ , ${\log _a}{a^n} = n$ , ${\log _a}a = 1$ and ${\log _a}1 = 0$ , we can use these efficiently during solving the questions.