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How do you solve \[{\log _6}\left( x \right) - {\log _6}\left( {x - 6} \right) = 1\] ?

Last updated date: 03rd Mar 2024
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IVSAT 2024
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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.
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