Question

How do you solve $4{e^{7x}} = 10273$?

Answer 1

Hint: First we have to divide each term of the given equation by $4$ and simplify. For this, divide each term in $4{e^{7x}} = 10273$ by $4$. Next, take the natural logarithm of both sides of the equation to remove the variable from the exponent. Next, expand the left side. For this, expand $\ln \left( {{e^{7x}}} \right)$ by moving $7x$ outside the logarithm. Next, divide each term by $7$ and simplify.
For this, divide each term in $7x = \ln \left( {\dfrac{{10273}}{4}} \right)$ by $7$. Then, we will get the solution of the given equation.

Formula used:
$\ln \left( {{a^m}} \right) = m\ln \left( a \right)$
$\ln \left( e \right) = 1$

Complete step by step solution:
Given equation: $4{e^{7x}} = 10273$
We have to find all possible values of $x$ satisfying a given equation.
So, first we have to divide each term by $4$ and simplify.
For this, divide each term in $4{e^{7x}} = 10273$ by $4$.
$ \Rightarrow \dfrac{{4{e^{7x}}}}{4} = \dfrac{{10273}}{4}$
Now, cancel the common factor of $4$.
$ \Rightarrow {e^{7x}} = \dfrac{{10273}}{4}$
Next, take the natural logarithm of both sides of the equation to remove the variable from the exponent.
$\ln \left( {{e^{7x}}} \right) = \ln \left( {\dfrac{{10273}}{4}} \right)$
Now, expand the left side.
For this, expand $\ln \left( {{e^{7x}}} \right)$ by moving $7x$ outside the logarithm as $\ln \left( {{a^m}} \right) = m\ln \left( a \right)$.
$ \Rightarrow 7x\ln \left( e \right) = \ln \left( {\dfrac{{10273}}{4}} \right)$
We know that the natural logarithm of $e$ is $1$.
$ \Rightarrow 7x \times 1 = \ln \left( {\dfrac{{10273}}{4}} \right)$
Multiply $7x$ by $1$, we get
$ \Rightarrow 7x = \ln \left( {\dfrac{{10273}}{4}} \right)$
Now, divide each term by $7$ and simplify.
For this, divide each term in $7x = \ln \left( {\dfrac{{10273}}{4}} \right)$ by $7$.
$ \Rightarrow \dfrac{{7x}}{7} = \dfrac{{\ln \left( {\dfrac{{10273}}{4}} \right)}}{7}$
Cancel the common factor of $7$.
$ \Rightarrow x = \dfrac{{\ln \left( {\dfrac{{10273}}{4}} \right)}}{7}$
The result can be shown in multiple forms.
Exact Form: $x = \dfrac{{\ln \left( {\dfrac{{10273}}{4}} \right)}}{7}$
Decimal Form: $x = 1.121568573$

Final solution: Hence, $x = \dfrac{{\ln \left( {\dfrac{{10273}}{4}} \right)}}{7}$ or $x = 1.121568573$ is the solution of $4{e^{7x}} = 10273$.

Note: In above question, we can find the solutions of given equation by plotting the equation, $4{e^{7x}} = 10273$ on graph paper and determine all its solutions.

From the graph paper, we can see that $x = 1.121568573$ is the solution of $4{e^{7x}} = 10273$.
Final solution: Hence, $x = \dfrac{{\ln \left( {\dfrac{{10273}}{4}} \right)}}{7}$ or $x = 1.121568573$ is the solution of $4{e^{7x}} = 10273$.