Question

The value of \[x\] obtained from the equation \[{\left( 4 \right)^{{{\log }_9}3}} + {\left( 9 \right)^{{{\log }_2}4}} = {\left( {10} \right)^{{{\log }_x}83}}\] will be \[2p\] . Find the value of \[p\] .

Answer 1

Hint: In order to solve the above question, we will use a few formulas concerning logarithm. One important formula of logarithm is \[\log {x^a} = a\log x\] . We will be using this formula to find the correct answer to the question. Another formula that we will be using is \[{\log _e}e = 1\] . These formulas will help you in solving the question easily.

Formula used: To solve this question, we will be using a few formulas of logarithm. These formulas are:
\[\log {x^a} = a\log x\] .
\[{\log _e}e = 1\] .

Complete step by step solution:
We are given that \[{\left( 4 \right)^{{{\log }_9}3}} + {\left( 9 \right)^{{{\log }_2}4}} = {\left( {10} \right)^{{{\log }_x}83}}\] .
We will solve the left-hand side first.
We have \[{\left( 4 \right)^{{{\log }_9}3}} + {\left( 9 \right)^{{{\log }_2}4}}\] ,
This can be written as \[{\left( 4 \right)^{{{\log }_3}{3^{\dfrac{1}{2}}}}} + {\left( 9 \right)^{{{\log }_2}{2^2}}}\] ,
Now we will use the 1) formula written above,
The equation will become,
\[{\left( 4 \right)^{\dfrac{1}{2}{{\log }_3}3}} + {\left( 9 \right)^{2{{\log }_2}2}}\]
\[ \Rightarrow {\left( 2 \right)^{2 \times \dfrac{1}{2}{{\log }_3}3}} + {\left( 9 \right)^{2{{\log }_2}2}}\]
In the next step, we will use the 2) formula mentioned above.
We get,
\[
  2 + {\left( 9 \right)^2} \\
   \Rightarrow 2 + 81 \\
   \Rightarrow 83 \\
 \]
From this we get that the left-hand side equals to \[83\] .
Now we will equate it with the right-hand side, we get,
\[83 = {\left( {10} \right)^{{{\log }_x}83}}\]
From this we find that \[x = 10\] .
Now we know that this value of \[x = 2p\] . To calculate the value of \[p\] :
\[\dfrac{x}{2} = p\]
Now the value of \[x = 10\]
\[
  \dfrac{{10}}{2} = p \\
   \Rightarrow 5 = p \\
 \] .
So, the correct answer is \[p = 5\] .

Note: To solve sums similar to the above questions, you need to always remember the formulas used to solve questions relating to logarithm. These formulas are very important and they help in making the question simple and easy to solve. To solve the above question, we have used two such formulas.