Question

How do you evaluate ${\log _{81}}3$.

Answer 1

Hint: The question belongs to the concept of logarithmic function. We will use the base change formula of logarithmic function and some properties of the logarithmic function. According to the base change formula, the base of a logarithmic function can be changed as ${\log _y}x = \dfrac{1}{{{{\log }_x}y}}$. We can also change the base of a logarithmic function to any positive real number that does not belong to 1, ${\log _a}x = \dfrac{{{{\log }_c}x}}{{{{\log }_c}a}}$, where $a,b$ and $x$ are all positive real numbers and $\left( {a,b} \right) \ne 1$. Logarithmic functions are the inverse of exponential functions. The inverse of the exponential function $y = {m^x}$ is $x = {m^y}$. The domain of a logarithmic function is real numbers greater than zero and the range of the logarithmic function is real numbers. A natural logarithmic function is defined as a logarithmic function with base. The logarithmic function with base ten is known as the common logarithmic function. The graph of a logarithmic function $y = {\log _a}x$ is equivalent to the graph of $y = {a^x}$ with respect to a reference line $y = x$. This relationship holds true for all any function and its inverse.

Complete step by step solution:
Step: 1 the given logarithmic function is,
${\log _{81}}3$
Use the base change formula to solve the logarithmic function.
The base change formula is,
${\log _a}b = \dfrac{{{{\log }_c}b}}{{{{\log }_c}a}}$
Where $a,b,c > 0$and$b,c \ne 1$.
Therefore,
$
   \Rightarrow {\log _{81}}3 = \dfrac{{{{\log }_3}3}}{{{{\log }_3}81}} \\
   \Rightarrow {\log _{81}}3 = \dfrac{{{{\log }_3}3}}{{{{\log }_3}{3^4}}} \\
 $
Step: 2 use the identity of the logarithmic function to evaluate the function.
Use the power change formula of logarithmic function.
${\log _a}{x^m} = m{\log _a}x$
Therefore,
$
   \Rightarrow {\log _{81}}3 = \dfrac{{{{\log }_3}3}}{{{{\log }_3}{3^4}}} \\
   \Rightarrow {\log _{81}}9 = \dfrac{{{{\log }_3}3}}{{4{{\log }_3}3}} \\
   \Rightarrow {\log _{81}}3 = \dfrac{1}{4} \\
 $

Final Answer:
Therefore, ${\log _{81}}3 = \dfrac{1}{4}$.

Note:
Use the base change formula of logarithmic function to evaluate the function. Consider the basic identity of logarithmic function and use them to evaluate the question. the base change formula of a logarithmic function ${\log _y}x = \dfrac{1}{{{{\log }_x}y}}$ can be used to determine the value of logarithmic function.