Question

How do you solve $\dfrac{1}{2}={{10}^{1.5x}}$?

Answer 1

Hint: In this problem we have to solve the given exponential equation which is $\dfrac{1}{2}={{10}^{1.5x}}$. We can clearly observe that we can’t write the left-hand side part which is $\dfrac{1}{2}$ as a power of $10$. So, we are going to apply the $\log $ function on both sides of the equation. Now we will use $\log \left( \dfrac{a}{b} \right)=\log a-\log b$ , $\log {{a}^{b}}=b\log a$ formulas in the obtained equation and simplify the equation. Now we will substitute the known values like $\log 1=0$, ${{\log }_{10}}10=1$ in the obtained equation. After that we will simplify the equation to get the required result.

Complete step by step solution:
Given the equation, $\dfrac{1}{2}={{10}^{1.5x}}$.
Applying $\log $ function on both sides of the above equation, then we will get
$\Rightarrow \log \left( \dfrac{1}{2} \right)=\log \left( {{10}^{1.5x}} \right)$
Applying the logarithmic formula $\log \left( \dfrac{a}{b} \right)=\log a-\log b$ in LHS of the above equation, then we will get
$\Rightarrow \log 1-\log 2=\log \left( {{10}^{1.5x}} \right)$
Now applying the logarithmic formula $\log {{a}^{b}}=b\log a$ in RHS of the above equation, then we can write
$\Rightarrow \log 1-\log 2=1.5x{{\log }_{10}}10$
We know that ${{\log }_{a}}a=1\Rightarrow {{\log }_{10}}10=1$. Substituting this value in the above equation, then we will get
$\Rightarrow \log 1-\log 2=1.5x$
We have the value $\log 1=0$, substituting this value in the above equation, then the above equation is modified as
$\begin{align}
  & \Rightarrow 0-\log 2=1.5x \\
 & \Rightarrow 1.5x=-\log 2 \\
\end{align}$
In the above equation we can observe that $1.5$ is in multiplication, so dividing the above equation with $1.5$ on both sides of the equation, then we will get
$\begin{align}
  & \Rightarrow \dfrac{1.5x}{1.5}=-\dfrac{\log 2}{1.5} \\
 & \Rightarrow x=-\dfrac{\log 2}{1.5} \\
\end{align}$
We can write $\dfrac{1}{1.5}=\dfrac{2}{3}$ in the above equation, then we will have
$x=-\dfrac{2}{3}\log 2$

Hence the solution of the given equation $\dfrac{1}{2}={{10}^{1.5x}}$ is $x=-\dfrac{2}{3}\log 2$.

Note: If you want to calculate the exact value of $x$, then we need to substitute the value ${{\log }_{10}}2=0.3010$ in the above equation and simplify it, then we will get
$\begin{align}
  & \Rightarrow x=-\dfrac{2\times 0.3010}{3} \\
 & \Rightarrow x=-0.2006 \\
\end{align}$.