Question

How do you solve ${3^x} = 243$?

Answer 1

Hint: Here we must know that when we are given such problems we can convert $243$ into the form of multiplication of its factors and then we will come to know that when $3$ is multiplied $5$ times it gives us $243$ and therefore we get the result as $5$.

Complete step by step solution:
Here we need to solve for the value of $x$ in the given equation which is ${3^x} = 243$
Here we need to know what the power of any term represents. This can be made clear with one example: If we have the term ${2^3}$ this means that $2$ is multiplied with itself $3$ times and hence its value will be $\left( 2 \right)\left( 2 \right)\left( 2 \right) = 8$ and therefore we can say that ${2^3} = 8$
Similarly we are given the term ${3^x}$ and therefore from the above example we can understand what it means. We can come to know from ${3^x}$ that here $3$ is multiplied with itself $x$ times and therefore we can say that it is actually $3 \times 3 \times 3 \times 3 \times 3...........x{\text{ times}}$
Hence we are actually given to find the number of times $3$ needs to be multiplied with itself in order to give us the result $243$
Hence we can find the factors of $243$ and we will get:
$
  243 = \left( 3 \right)\left( {81} \right) \\
  81 = \left( 3 \right)\left( {27} \right) \\
  27 = \left( 3 \right)\left( 9 \right) \\
  9 = \left( 3 \right)\left( 3 \right) \\
  3 = \left( 3 \right)\left( 1 \right) \\
 $
Hence we can say that:
$243 = 3 \times 3 \times 3 \times 3 \times 3$

Hence we can say that when $3$ is multiplied $5$ times we get the result as $243$
Hence we can say that $x = 5$

Note:
This question can also be converted in the form where we can be given to find $x$ in ${243^x} = 3$.
And here we can say that $x = \dfrac{1}{5}$.
So we must know that if ${a^x} = b$ then we can also write that $a = {b^{\dfrac{1}{x}}}$.