Question

How do you solve: $3\log 2x=4$?

Answer 1

Hint: To solve the above logarithmic equation we first of all going to divide both the sides by 3 then we are going to use the following logarithm property which states that: ${{\log }_{b}}a=c$ then the relation between $a,b\And c$ is $a={{b}^{c}}$. And in this way, we can get the value of x.

Complete step-by-step solution:
The logarithmic expression given in the above problem is as follows:
$3\log 2x=4$
We are asked to find the solution to the above equation which we are going to solve by dividing 3 on both the sides of the above equation and we get,
$\dfrac{3}{3}\log 2x=\dfrac{4}{3}$
In the above equation, 3 will get cancelled out from the numerator and the denominator of the L.H.S of the above equation and we get,
$\log 2x=\dfrac{4}{3}$
If not given then we have to take the base of the above logarithm as 10 so rewriting the above equation we get,
${{\log }_{10}}2x=\dfrac{4}{3}$
Now, we are going to use the following logarithm property to solve the above equation as follows:
${{\log }_{b}}a=c$
Then the relation between x, y and z is as follows:
$a={{b}^{c}}$
Substituting $a=2x,b=10,c=\dfrac{4}{3}$ in the above equation we get,
$2x={{\left( 10 \right)}^{\dfrac{4}{3}}}$
Dividing 2 on both the sides of the above equation we get,
$\dfrac{2x}{2}=\dfrac{1}{2}{{\left( 10 \right)}^{\dfrac{4}{3}}}$
In the above equation, in the L.H.S of the above equation, 2 will get cancelled out from the numerator and the denominator and we get,
$x=\dfrac{1}{2}{{\left( 10 \right)}^{\dfrac{4}{3}}}$
Hence, we have found the solution of the given equation in x.

Note: We can check whether the given value of x is correct or not by substituting that value of x in the above equation and see if that value is satisfying the given equation or not.
The value of $x$ which we are getting in the above solution is $\dfrac{1}{2}{{\left( 10 \right)}^{\dfrac{4}{3}}}$ and the equation given above is as follows:
$3\log 2x=4$
Substituting $x=\dfrac{1}{2}{{\left( 10 \right)}^{\dfrac{4}{3}}}$ in the above equation we get,
$3\log 2\left( \dfrac{1}{2}{{\left( 10 \right)}^{\dfrac{4}{3}}} \right)=4$
In the L.H.S of the above equation, in the logarithm, 2 will get cancelled out from the numerator and the denominator and we get,
$3\log \left( {{\left( 10 \right)}^{\dfrac{4}{3}}} \right)=4$ ………(1)
Now, we are going to use the following property of logarithm which states that:
$\log {{x}^{a}}=a\log x$
Substituting $x=10$ and $a=\dfrac{4}{3}$ in the above equation and we get,
$\log {{10}^{\dfrac{4}{3}}}=\dfrac{4}{3}\log 10$
We know that if not given then base of the logarithm is taken as 10 so taking the base as 10 in the above logarithm we get,
$\begin{align}
  & \log {{10}^{\dfrac{4}{3}}}=\dfrac{4}{3}{{\log }_{10}}10 \\
 & \Rightarrow \log {{10}^{\dfrac{4}{3}}}=\dfrac{4}{3}\left( 1 \right) \\
 & \Rightarrow \log {{10}^{\dfrac{4}{3}}}=\dfrac{4}{3} \\
\end{align}$
Using above relation in eq. (1) we get,
$\begin{align}
  & 3\log \left( {{\left( 10 \right)}^{\dfrac{4}{3}}} \right)=4 \\
 & \Rightarrow 3\left( \dfrac{4}{3} \right)=4 \\
 & \Rightarrow 4=4 \\
\end{align}$
In the above equation, L.H.S = R.H.S so the value of x which we have found in the above solution is correct.