Question

Compute the value of ${81^{\dfrac{1}{{{{\log }_5}3}}}} + {27^{{{\log }_9}36}} + {3^{\dfrac{4}{{{{\log }_7}9}}}}$.

Answer 1

Hint: In the above question, we are required to find the value of the expression involving logarithms and exponents. So to solve this question, we must know what logarithm functions are. A logarithm function is the inverse of an exponential function and is of the form $y = {\log _x}a$. We will use properties of logarithms such as ${\log _x}a = \dfrac{1}{{{{\log }_a}x}}$ and \[{y^{{{\log }_x}a}} = {a^{{{\log }_x}y}}\] to get to the required answer.

Complete step by step answer:
We have to find the value of the expression: ${81^{\dfrac{1}{{{{\log }_5}3}}}} + {27^{{{\log }_9}36}} + {3^{\dfrac{4}{{{{\log }_7}9}}}}$.
Now, we know the logarithmic property ${\log _x}a = \dfrac{1}{{{{\log }_a}x}}$.
So, we have, \[\dfrac{1}{{{{\log }_5}3}} = {\log _3}5\] and \[\dfrac{1}{{{{\log }_7}9}} = {\log _9}7\]. So, we get,
\[ \Rightarrow {81^{{{\log }_3}5}} + {27^{{{\log }_9}36}} + {3^{4{{\log }_9}7}}\]
Now, we know the logarithmic property \[{y^{{{\log }_x}a}} = {a^{{{\log }_x}y}}\]. So, we get,
\[ \Rightarrow {5^{{{\log }_3}81}} + {36^{{{\log }_9}27}} + {7^{4{{\log }_9}3}}\]
We also know that $81 = {3^4}$, $27 = {9^{\dfrac{3}{2}}}$ and $3 = {9^{\dfrac{1}{2}}}$. So, we get,
\[ \Rightarrow {5^{{{\log }_3}{3^4}}} + {36^{{{\log }_9}{9^{\dfrac{3}{2}}}}} + {7^{{{\log }_9}{9^{\dfrac{1}{2}}}}}\]
Now, using the logarithmic property \[\log {x^n} = n\log x\], we get,
\[ \Rightarrow {5^{4{{\log }_3}3}} + {36^{\dfrac{3}{2}{{\log }_9}9}} + {7^{\dfrac{4}{2}{{\log }_9}9}}\]
Now, we know that the value of \[{\log _a}a\] is one. So, we get,
\[ \Rightarrow {5^4} + {36^{\dfrac{3}{2}}} + {7^{\dfrac{4}{2}}}\]
Now, cancelling the common factors in numerator and denominator, we get,
\[ \Rightarrow {5^4} + {\left( {\sqrt {36} } \right)^3} + {7^2}\]
We know that square root of \[36\] is $6$. Then, we get,
\[ \Rightarrow {5^4} + {6^3} + {7^2}\]
Now, evaluating the powers of the numbers,
\[ \Rightarrow 625 + 216 + 49\]
Simplifying the calculations, we get,
\[ \Rightarrow 890\]
So, the value of the expression ${81^{\dfrac{1}{{{{\log }_5}3}}}} + {27^{{{\log }_9}36}} + {3^{\dfrac{4}{{{{\log }_7}9}}}}$ is \[890\].

Note: There are several laws of the logarithm that make the calculations easier and help us evaluate the logarithm functions. While applying the laws of the logarithm, we should keep in mind the base of the logarithm functions involved in all the calculations and steps. We should remember the powers of simple numbers to ask such questions. Take care of calculations so as to be sure of the final answer.