Answer

Verified

340.8k+ views

**Hint:**In the given question, we have been asked to find the value of ‘x’ and it is given that \[\ln \left( {{x}^{7}} \right)-\ln \left( {{x}^{2}} \right)=5\]. In order to find the value of ‘x’, first we will apply the quotient property of logarithm which states that \[{{\log }_{b}}m-{{\log }_{b}}n={{\log }_{b}}\left( \dfrac{m}{n} \right)\] . Then we need to apply the definition of logarithm, which states that \[\log \left( {{x}^{a}} \right)=a\log x\] and then simplify the equation further. After applying log formulae to the equation, we need to solve the equation in the way we solve general linear equations.

**Formula used:**

● The properties of logarithm to combine two natural logs;

Using the quotient property of logarithm, which states that \[{{\log }_{b}}m-{{\log }_{b}}n={{\log

}_{b}}\left( \dfrac{m}{n} \right)\]

● The definition of logarithm, says that \[\log \left( {{x}^{a}} \right)=a\log x\]

**Complete step by step solution:**

We have given that,

\[\ln \left( {{x}^{7}} \right)-\ln \left( {{x}^{2}} \right)=5\]

The properties of logarithm to combine two natural logs;

Using the quotient property of logarithm, i.e.

\[{{\log }_{b}}m-{{\log }_{b}}n={{\log }_{b}}\left( \dfrac{m}{n} \right)\]

Applying the property in the above equation, we get

\[\Rightarrow \ln \left( \dfrac{{{x}^{7}}}{{{x}^{2}}} \right)=5\]

On simplifying the above equation, we get

\[\Rightarrow \ln \left( {{x}^{5}} \right)=5\]

By the definition of logarithm, i.e.

\[\log \left( {{x}^{a}} \right)=a\log x\]

Using this, we get

\[\Rightarrow 5\ln \left( x \right)=5\]

Multiplying both the sides of the equation by 5, we get

\[\Rightarrow \dfrac{5\ln \left( x \right)}{5}=\dfrac{5}{5}\]

On simplifying the above, we get

\[\Rightarrow \ln \left( x \right)=1\]

Therefore,

\[\Rightarrow x=e\]

**Thus, the value of ‘x’ equals to ‘e’ is the required solution.**

**Note:**In the given question, we need to find the value of ‘x’. To solve these types of questions, we used the basic formulas of logarithm. Students should always be required to keep in mind all the formulae for solving the question easily. After applying log formulae to the equation, we need to solve the equation in the way we solve general linear equations.

Recently Updated Pages

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

How do you arrange NH4 + BF3 H2O C2H2 in increasing class 11 chemistry CBSE

Is H mCT and q mCT the same thing If so which is more class 11 chemistry CBSE

What are the possible quantum number for the last outermost class 11 chemistry CBSE

Is C2 paramagnetic or diamagnetic class 11 chemistry CBSE

What happens when entropy reaches maximum class 11 chemistry JEE_Main

Trending doubts

How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

Draw a diagram of nephron and explain its structur class 11 biology CBSE

What is the color of ferrous sulphate crystals? How does this color change after heating? Name the products formed on strongly heating ferrous sulphate crystals. What type of chemical reaction occurs in this type of change.

Write the difference between soap and detergent class 10 chemistry CBSE

Give 10 examples of unisexual and bisexual flowers

Differentiate between calcination and roasting class 11 chemistry CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

What is the difference between anaerobic aerobic respiration class 10 biology CBSE

a Why did Mendel choose pea plants for his experiments class 10 biology CBSE