Answer

Verified

341.1k+ views

**Hint**: Here the given function is a logarithm function it can be defined as logarithmic functions are the inverses of exponential functions. The given log function has base value 6 by using some of the Basic Properties of logarithmic functions. And by further simplification we get a required solution.

**Complete step by step solution:**

The function from positive real numbers to real numbers to real numbers is defined as \[{\log _b}:{R^ + } \to R \Rightarrow {\log _b}\left( x \right) = y\] , if \[{b^y} = x\] , is called logarithmic function or the logarithm function is the inverse form of exponential function.

There are some basic logarithms properties

1. product rule :- \[\log \left( {mn} \right) = \log m + \log n\]

2. Quotient rule :- \[\log \left( {\dfrac{m}{n}} \right) = \log m - \log n\]

3. Power rule :- \[\log \left( {{m^n}} \right) = n.\log m\]

Now, Consider the given logarithm function, it has base 6

\[ \Rightarrow {\log _6}\left( x \right) - {\log _6}\left( {x - 6} \right) = 1\] (1)

By using the quotient rule of logarithm properties equation (1) can be rewritten as

\[ \Rightarrow {\log _6}\left( {\dfrac{x}{{x - 6}}} \right) = 1\] (2)

By the definition of logarithm function \[ \Rightarrow {\log _b}\left( x \right) = y\] can be written as \[{b^y} = x\] .

Then, equation (2) becomes

\[ \Rightarrow \dfrac{x}{{x - 6}} = {6^1}\]

On simplification we get

\[ \Rightarrow \dfrac{x}{{x - 6}} = 6\]

Multiply both side by \[\left( {x - 6} \right)\] , we get

\[ \Rightarrow x = 6\left( {x - 6} \right)\]

Using distributive property on RHS then

\[ \Rightarrow x = 6x - 36\]

Isolate the x variable on one side of the equation, by subtracting 6x on both side, then

\[ \Rightarrow x - 6x = 6x - 36 - 6x\]

On simplification we get

\[ \Rightarrow - 5x = - 36\]

Cancel ‘-’ ve on both sides, then

\[ \Rightarrow 5x = 36\]

To solve x, divide both sides by 5.

\[\therefore x = \dfrac{{36}}{5}\]

Hence, the value of x in the function \[{\log _6}\left( x \right) - {\log _6}\left( {x - 6} \right) = 1\] is \[\dfrac{{36}}{5}\] .

**So, the correct answer is “ \[\dfrac{{36}}{5}\] ”.**

**Note**: The question contains the log terms we must know the logarithmic properties which are the standard properties. By applying properties we can solve the question in an easy manner. We apply the formula \[{\log _b}\left( x \right) = y\] that can be written as \[{b^y} = x\] . where it is necessary. Hence, we obtain the desired result.

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

Why is the adrenaline hormone called fight or flight class 11 biology CBSE

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

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Differentiate between lanthanoids and actinoids class 12 chemistry 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.

Give 10 examples of unisexual and bisexual flowers

Open circulatory system is present in I Arthropods class 12 biology CBSE

Name the highest peak of the Indian Himalayas class 8 social science CBSE