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.