Question

How do you solve \[{{\log }_{x}}16=4?\]

Answer 1

Hint: We are given an expression as \[{{\log }_{x}}16=4\] and we are asked to solve this. We will learn what does the solution of the problem means and then we will use \[{{\log }_{a}}b=c\] which means \[{{a}^{c}}=b.\] Using this we will solve further and then we will change 16 into exponent form and lastly we will compare the two exponents and find our solution. We will also use \[\log {{a}^{b}}=b\log a\] and \[{{\log }_{a}}\left( a \right)=1\] while verifying our solution.

Complete step by step answer:
We are given \[{{\log }_{a}}16=4\] and we have to find its solution. The solution in those values of x which when inserted in the equation will satisfy the given equation. To solve the given problem we need to know about how the log function behaves. We know that the log function is the inverse of the exponential function. We can always convert log into exponential and exponential to log function. We have that for \[{{\log }_{a}}b=c,\] we can write it in exponential as \[{{a}^{c}}=b.\] First, we will use this property of log to simplify our problem. We are given \[{{\log }_{x}}16=4,\] comparing it with \[{{\log }_{a}}b=c,\] we have a = x, b = 16 and c = 4. Now as \[{{\log }_{a}}b=c,\] we can write \[{{a}^{c}}=b.\] So, we will get
\[{{\log }_{x}}16=4\]
\[\Rightarrow {{x}^{4}}=16\]
Now we have our equation into exponential form. We will first convert 16 into exponential form as 16 is \[2\times 2\times 2\times 2.\] So, we can write
\[16=2\times 2\times 2\times 2\]
\[\Rightarrow 16={{2}^{4}}\]
So, we get,
\[\Rightarrow {{x}^{4}}={{2}^{4}}\]
Now we have an exponential equation whose power is the same. So, as they are equal it means the base must be equal. So, \[{{x}^{4}}={{2}^{4}}\] implies that x = 2.

So, we get x = 2 as the solution of \[{{\log }_{x}}16=4.\]

Note: We can verify the answer by putting x = 2 and checking that x = 2 satisfies the problem or not. Now, we put x = 2 in \[{{\log }_{x}}16=4,\] we get \[{{\log }_{2}}16=4.\] As \[16={{2}^{4}},\] so we get, \[{{\log }_{2}}\left( {{2}^{4}} \right)=4.\] Now, we know \[{{\log }_{a}}{{a}^{b}}=b\log a.\] So, we get,
\[{{\log }_{2}}{{2}^{4}}=4{{\log }_{2}}2\]
\[\Rightarrow 4{{\log }_{2}}\left( 2 \right)=4\]
Now, as \[{{\log }_{a}}\left( a \right)=1,\] so we get \[{{\log }_{2}}\left( 2 \right)=1.\] Hence \[4{{\log }_{2}}\left( 2 \right)=4,\] we get 4 = 4. Hence satisfied. Therefore, x = 2 is our solution.