Question

How do you solve ${{2}^{3x}}=32$?

Answer 1

Hint: We will factorize the number 32 and find its prime factors. We will write the number 32 as a power of 2. Then we will equate the powers of 2 on the left hand side and the right hand side of the equation. We will get a linear equation in one variable. Then we will simplify this obtained equation to get the value of the variable.

Complete step-by-step solution:
The given equation is ${{2}^{3x}}=32$. Let us look at the prime factorization of the number 32. It is as follows,
$\begin{align}
  & 2\left| \!{\underline {\,
  32 \,}} \right. \\
 & 2\left| \!{\underline {\,
  16 \,}} \right. \\
 & 2\left| \!{\underline {\,
  8 \,}} \right. \\
 & 2\left| \!{\underline {\,
  4 \,}} \right. \\
 & 2\left| \!{\underline {\,
  2 \,}} \right. \\
 & \left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}$
The prime factorization shows us that we have $32={{2}^{5}}$. Substituting this value of 32 in the given equation, we get the following,
${{2}^{3x}}={{2}^{5}}$
The base of the exponential functions on the left hand side and the right hand side of the above equation is 2. So, to get the same value by taking the powers of two, the powers of 2 on both sides of the equation must be equal. Equating the powers of 2 on both sides of the equation, we get the following,
$3x=5$
We have a linear equation with one variable. To solve this equation, we will divide both sides of the equation by 3. So, we have the following,
$x=\dfrac{5}{3}$
Thus, we have obtained the solution of the given equation.

Note: We should be familiar with the exponential function, its properties and rules for simplification. After converting 32 into a power of 2, we can take the logarithm on both sides. Using the logarithm power rule, we get the equation $3x\times \log 2=5\times \log 2$. Cancelling the log 2 from both sides of the equation, we obtain the same linear equation as above. So, using this alternate method, we will get the same solution.