The value of $x$ obtained from equation \[{(4)^{{{\log }_9}}}^3 + {(9)^{{{\log }_2}}}^4 = {(10)^{{{\log }_x}}}^{83}\] will be 2p. Find the value of p.
A) 0
B) 1
C) 2
D) 3
E) 4
F) 5
G) 6
H) 7
I) 8
J) 9

Hint: The logarithmic function is the inverse function of the exponential function given by the formula \[{\log _b}a = c \Leftrightarrow {b^c} = \log a\], where b is the base of the logarithmic function. The logarithm is the mathematical operation that tells how many times a number or base is multiplied by itself to reach another number. There are five basic properties of the logarithm, namely Product rule, Quotient rule, Change of base rule, power rule, and equality rule.
The power rule of the logarithm is basically used to simplify the logarithm of power, rewriting it as the product of the exponent time to the logarithm base given by the formula \[{\log _a}{y^b} = b{\log _a}y\].
In this question, we have to evaluate the value of the given logarithmic function by using the properties.

Complete step-by-step solution
Using the formula of the logarithmic function \[{\log _a}{y^b} = b{\log _a}y\] in the given equation as
\[
\Rightarrow {(4)^{{{\log }_9}}}^3 + {(9)^{{{\log }_2}}}^4 = {(10)^{{{\log }_x}}}^{83} \\
\Rightarrow {4^{\left( {\dfrac{{\log 3}}{{\log 9}}} \right)}} + {9^{\left( {\dfrac{{\log 4}}{{\log 2}}} \right)}} = {10^{{{\log }_x}}}^{83} \\
\Rightarrow {4^{\left( {\dfrac{{\log 3}}{{\log {3^2}}}} \right)}} + {9^{\left( {\dfrac{{\log {2^2}}}{{\log 2}}} \right)}} = {10^{{{\log }_x}}}^{83} - - - - (i) \\
 \]
Now, using the formula of the logarithmic function $\log {a^b} = b\log a$ in equation (i) as
\[
\Rightarrow {4^{\left( {\dfrac{{\log 3}}{{2\log 3}}} \right)}} + {9^{\left( {\dfrac{{2\log 2}}{{\log 2}}} \right)}} = {10^{{{\log }_x}}}^{83} \\
\Rightarrow {4^{\left( {\dfrac{1}{2}} \right)}} + {9^2} = {10^{{{\log }_x}}}^{83} \\
\Rightarrow 2 + 81 = {10^{{{\log }_x}}}^{83} \\
\Rightarrow {10^{{{\log }_x}}}^{83} = 83 - - - - (ii) \\
 \]
Taking logarithmic function both sides of the equation (ii) so that the raised power can come as in multiplication:
\[
\Rightarrow \log \left( {{{10}^{{{\log }_x}}}^{83}} \right) = \log 83 \\
\Rightarrow {\log _x}83 \times \log 10 = \log 83 \\
\Rightarrow \dfrac{{\log 83}}{{\log x}} = \log 83 \\
\Rightarrow \log x = 1 \\
  x = 10 \\
 \]
According to question,
\[
\Rightarrow {(4)^{{{\log }_9}}}^3 + {(9)^{{{\log }_2}}}^4 = {(10)^{{{\log }_x}}}^{83} = 2p \\
\Rightarrow 10 = 2p \\
  p = 5 \\
 \]
Hence, the value of ‘p’ in the expression \[{(4)^{{{\log }_9}}}^3 + {(9)^{{{\log }_2}}}^4 = {(10)^{{{\log }_x}}}^{83} = 2p\] is 5.

Hence Option F is the correct answer.

Note: Always remember the standard properties of the logarithmic functions. Product rule, quotient rule and the power rule are some of the most commonly and widely used logarithmic properties.
Some of the useful properties of the logarithmic functions are:
$
  {\log _a}xy = {\log _a}x + {\log _a}y \\
  {\log _a}\dfrac{x}{y} = {\log _a}x - {\log _a}y \\
  {\log _a}{x^n} = n{\log _a}x \\
  {\log _a}a = 1 \\
 $