Question

How do I find ${\log _{27}}9$ $?$

Answer 1

Hint : Given a logarithm of the form ${\log _b}x$ , to find the value rewrite the given function in exponential form using the definition of logarithm. If x and b are positive real numbers and b does not equal 1, then ${\log _b}x = y$ is equivalent to ${b^y} = x$ . Further simplification you get the required value.

Complete step-by-step answer :
The logarithm is the inverse function to exponentiation. That means the logarithm of a given number x is the exponent to which another fixed number, the base b, must be raised, to produce that number x. The logarithm function expressed as follows
${\log _b}\left( x \right) = y$ exactly if ${b^y} = x$ and $x > 0$ and $b > 0$ and $b \ne 1$ else if $b = 1$ the value will be not defined
The log function with base $b = 10$ is called the common logarithmic function and it is denoted by ${\log _{10}}$ or simply written as $\log$ .
The log function having base $e$ is called the natural logarithmic function and it is denoted by ${\log _e}$ .
Consider the given function
$\Rightarrow \,\,{\log _{27}}9$
Here, the logarithm function log having base 27
The given expression can be written as
$\Rightarrow \,\,{\log _{27}}9 = x$
Rewrite ${\log _{27}}9 = x$ in exponential form using the definition of logarithm, then
$\Rightarrow \,\,{27^x} = 9$
Create expressions in the equation that all have equal bases.
$\Rightarrow \,\,{\left( {{3^3}} \right)^x} = {3^2}$
Using the rule of law of indices i.e., $\,{\left( {{x^m}} \right)^n} = {x^{mn}}$
$\Rightarrow \,\,{3^{3x}} = {3^2}$
Since the bases are the same, then two expressions are only equal if the exponents are also equal, then
$\Rightarrow \,\,\,3x = 2$
To solve x
$\therefore \,\,\,\,x = \dfrac{2}{3}$
Hence, the value of ${\log _{27}}9$ is $\dfrac{2}{3}$ or $0.6$
So, the correct answer is “ $\dfrac{2}{3}$ or $0.6$ ”.

Note : To solve the logarithmic equation we need to convert the equation to the exponential form. The exponential form of a number is defined as the number of times the number is multiplied by itself. The general form of logarithmic equation is ${\log _b}x = y$ and it is converted to exponential form as $x = {b^y}$ . And we obtained the value of x. Hence we obtain the result or solution for the equation.