Question

How do you solve \[{{3}^{x}}=6561\]?

Answer 1

Hint: This question belongs to the topic of algebra. In solving this question, we will first do the prime factorization of 6561. After getting the factorization, we will balance the equation so that the bases of both sides of the equation are equal. After that, we will use some formulas and solve the further equation and find the value of x.

Complete step by step solution:
Let us solve this question.
In this question, we have asked to solve the term \[{{3}^{x}}=6561\]. Or, we can say that we have to solve or find the value of x from the equation \[{{3}^{x}}=6561\].
The equation we have to solve is
\[{{3}^{x}}=6561\]
Let us first factorize the term 6561.
\[\begin{align}
  & 3\left| \!{\underline {\,
  6561 \,}} \right. \\
 & 3\left| \!{\underline {\,
  2187 \,}} \right. \\
 & 3\left| \!{\underline {\,
  729 \,}} \right. \\
 & 3\left| \!{\underline {\,
  243 \,}} \right. \\
 & 3\left| \!{\underline {\,
  81 \,}} \right. \\
 & 3\left| \!{\underline {\,
  27 \,}} \right. \\
 & 3\left| \!{\underline {\,
  9 \,}} \right. \\
 & 3\left| \!{\underline {\,
  3 \,}} \right. \\
 & 1\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}\]
So, from the above factorization we can write
\[6561=3\times 3\times 3\times 3\times 3\times 3\times 3\times 3\]
The equation \[{{3}^{x}}=6561\] can also be written as
\[\Rightarrow {{3}^{x}}=3\times 3\times 3\times 3\times 3\times 3\times 3\times 3\]
As we know that if base are same and they are multiplied then powers can be added to the base after removing the multiplied numbers or we can say that \[{{x}^{a}}\times {{x}^{b}}={{x}^{\left( a+b \right)}}\]
So, the above equation can be written as
\[\Rightarrow {{3}^{x}}={{3}^{1}}\times {{3}^{1}}\times {{3}^{1}}\times {{3}^{1}}\times {{3}^{1}}\times {{3}^{1}}\times {{3}^{1}}\times {{3}^{1}}\]
The above equation can also be written as
\[\Rightarrow {{3}^{x}}={{3}^{\left( 1+1+1+1+1+1+1+1 \right)}}\]
The above equation can also be written as
\[\Rightarrow {{3}^{x}}={{3}^{8}}\]
As we know that if base are same and they are equal, then their powers can also be equal, so we can write
\[\Rightarrow x=8\]
Now, we have solved the equation \[{{3}^{x}}=6561\] and have got the value of x as 8.

Note: As we can see that this question is from the topic of algebra, so we should have a better knowledge in the topic of algebra. We should know how to find the factorization of any number.
Remember that whenever we have to multiply the two numbers having same base but different powers, then the multiplication can be written as the base but with power as sum of different powers that we have take initially. We can understand this from the following formula:
\[{{x}^{a}}\times {{x}^{b}}={{x}^{\left( a+b \right)}}\]
Here, we can see that x is the base and ‘a’ and ‘b’ are the powers and they are added to the power of the base x which we can see in the right side of the equation.
And, always remember that if the equation is written in the form of
\[{{x}^{a}}={{x}^{b}}\]
Here, we can see that the bases are equal to the both side of the equation. Then, we can say that \[a=b\]