Question

How do you solve the algebraic expression ${{8}^{5x}}={{16}^{3x+4}}$

Answer 1

Hint: In order to solve the given equation first we need to make the base on both sides to equal. For this we can rewrite 8 and 16 in the form of base 2 with power 3 and 4 respectively. Then we will compare the powers of both sides and form an equation then simplify the obtained equation to get the desired answer.

Complete step-by-step solution:
We have been given that ${{8}^{5x}}={{16}^{3x+4}}$.
We have to solve the given expression.
Now, we know that $8={{2}^{3}}$ and $16={{2}^{4}}$
Now, substituting the values in the equation and rewrite the equation we will get
$\Rightarrow {{\left( {{2}^{3}} \right)}^{5x}}={{\left( {{2}^{4}} \right)}^{3x+4}}$
Now we know that by property of exponentiation we have ${{\left( {{x}^{m}} \right)}^{n}}={{x}^{mn}}$
Applying the property to the above obtained equation we will get
\[\Rightarrow {{\left( 2 \right)}^{3\times 5x}}={{\left( 2 \right)}^{4\left( 3x+4 \right)}}\]
Now, as the bases are same on both sides of the equation so when we compare the powers we will get
\[\Rightarrow 15x=4\left( 3x+4 \right)\]
Simplifying the above obtained equation we will get
$\begin{align}
  & \Rightarrow 15x=12x+16 \\
 & \Rightarrow 15x-12x=16 \\
 & \Rightarrow 3x=16 \\
 & \Rightarrow x=\dfrac{16}{3} \\
\end{align}$
Hence on solving the given expression we get the value of x as $\dfrac{16}{3}$.

Note: We can also solve the given expression by taking the logarithm after converting the equation with the same bases. We can take logarithm with base 2 as we know that ${{\log }_{a}}a=1$ . Then we get the equation \[15x=4\left( 3x+4 \right)\]. By simplifying the obtained equation we will get the value of x. We can also verify the answer by putting the value of x obtained in the given expression.