Question

How do you solve \[8\ln (3x) = 48\]?

Answer 1

Hint: Here in this question, we have to solve the given equation to the x variable. The given equation is the logarithmic equation with one variable x, this can be solve by using some logarithmic property and add or subtract the necessary term from each side of the equation to isolate the term with the variable x, then multiply or divide each side of the equation by the appropriate value, while keeping the equation balanced then solve the resultant balance equation for the x value.

Complete step-by-step solution:
Consider the given equation
\[8\ln (3x) = 48\]--------(1)
Where x is the variable
Now, we have to solve the above logarithmic equation for the variable x
Divide 8 on both side of equation (1), then
\[ \Rightarrow \,\,\,\dfrac{8}{8}\ln (3x) = \dfrac{{48}}{8}\]
On simplification, we get
\[ \Rightarrow \,\,\,\ln (3x) = 6\]--------(2)
Take the inverse function (or place both sides as an exponent to e). This will cancel the using one of the logarithmic property i.e., \[{e^{\ln \left( x \right)}} = x\]
Equation (2) can be written as
\[ \Rightarrow \,\,\,{e^{\ln (3x)}} = {e^6}\]
By the logarithmic property
\[ \Rightarrow \,\,\,3x = {e^6}\]
We want to solve the equation for x, then we have divide both side of equation by 3, then
\[ \Rightarrow \,\,\,\dfrac{3}{3}x = \dfrac{{{e^6}}}{3}\]
On simplification, we get
\[ \Rightarrow \,\,\,x = \dfrac{{{e^6}}}{3}\]
As we know the approximate value of e is 2.718, on substituting the value, then
\[ \Rightarrow \,\,\,x = \dfrac{{{{\left( {2.718} \right)}^6}}}{3} \cong 134.39\]

Hence, the required solution is \[\,x = \dfrac{{{e^6}}}{3} \cong 134.39\].

Note: We solve the given equation for the variable x. As we know that the logarithmic function and exponential function are inverse of each other. So we take antilog on both sides and we simplify the given term. On simplification we use the simple arithmetic operations to the equation and hence we obtain the solution for the question.