Question

How do you condense \[\ln 8+2\ln 5-\ln 10\]?

Answer 1

Hint: This question is from the topic of logarithms. In this, we are going to solve the term given in the question. In solving this question, we will first see the formulas of logarithms. After that, we will apply those formulas in solving this question. After solving the question using those formulas, we will get our answer.

Complete step by step answer:
Let us solve this question.
In this question, we have asked to condense the term which is given in the question. Or, we can say that we have to solve the given term. The term which we have to solve and given in the question is:
\[\ln 8+2\ln 5-\ln 10\]
Let us know the formulas of logarithms. The formulas for logarithms are in the following:
\[\ln {{a}^{b}}=b\ln a\]
If bases of the logarithms are same, then we can write
\[\ln a+\ln b=\ln \left( ab \right)\]
\[\ln a-\ln b=\ln \left( \dfrac{a}{b} \right)\]
In the above formula, we can see that the bases are same. (The base of \[\ln \] is ‘e’. The term \[\ln \] can also be written as \[\ln ={{\log }_{e}}\].)
Now, using the formula of logarithms that is \[\ln {{a}^{b}}=b\ln a\], we can write the term \[\ln 8+2\ln 5-\ln 10\] as
\[\ln 8+2\ln 5-\ln 10=\ln 8+\ln {{5}^{2}}-\ln 10\]
The above equation can also be written as
\[\Rightarrow \ln 8+2\ln 5-\ln 10=\ln 8+\ln 25-\ln 10\]
Now, using the formula of logarithms that is \[\ln a+\ln b=\ln \left( ab \right)\], we can write the above equation as
\[\Rightarrow \ln 8+2\ln 5-\ln 10=\ln \left( 8\times 25 \right)-\ln 10\]
As we know that 8 multiplied by 25 is 200, so we can write the above equation as
\[\Rightarrow \ln 8+2\ln 5-\ln 10=\ln 200-\ln 10\]
Using the formulas of logarithms that is \[\ln a-\ln b=\ln \left( \dfrac{a}{b} \right)\], we can write the above equation as
\[\Rightarrow \ln 8+2\ln 5-\ln 10=\ln \left( \dfrac{200}{10} \right)\]
The above equation can also be written as
\[\Rightarrow \ln 8+2\ln 5-\ln 10=\ln 20\]

Now, we can say that the condensed equation of the equation \[\ln 8+2\ln 5-\ln 10\] is \[\ln 20\].

Note: We should have a better knowledge in the topic of logarithms to solve this type of question easily. Remember that if bases of logarithms are the same then they can be added easily. For instance, we can see the following formulas:
\[\ln a+\ln b=\ln \left( ab \right)\]
\[\ln a-\ln b=\ln \left( \dfrac{a}{b} \right)\]
\[\ln {{a}^{b}}=b\ln a\]
\[\ln ={{\log }_{e}}\]
Remember the above formulas. They are very helpful in this type of question.