Question

How do you solve ${e^{2x}} = 50$?

Answer 1

To solve the given equation, take natural logarithm on both the sides of the equation to remove the variable from the exponent, as Logarithmic functions are the inverses of exponential functions hence by this, we can get the value of $x$.

Complete step by step solution:
Let us write the given equation
${e^{2x}} = 50$ ……………………………………. 1
To solve this equation, take natural logarithm on both the sides of the equation 1 i.e.,
$\ln \left( {{e^{2x}}} \right) = \ln \left( {50} \right)$ ……………………………….. 2
Expand the LHS part by moving $2x$ outside the logarithm of equation 2, hence we get
$2x\ln \left( e \right) = \ln \left( {50} \right)$
As we know the logarithm of function ‘$e$’ is 1, hence substituting this value in above equation
$2x \cdot 1 = \ln \left( {50} \right)$
In which the value of $\ln \left( e \right)$= 1.
$2x = \ln \left( {50} \right)$ ………………………………..… 3
As we need to find the value of $x$, now divide each term by 2 in equation 3 and simplifying the terms as
$\dfrac{{2x}}{2} = \dfrac{{\ln \left( {50} \right)}}{2}$
After simplifying we divide $x$ by 1 and we get,
$x = \dfrac{{\ln \left( {50} \right)}}{2}$ ………………………………… 4
On further simplification of the $x$ value in equation 4 i.e., finding the value of $\ln \left( {50} \right)$ and dividing it by 2 we get the value of $x$ as
$x = 1.956011$.
Therefore, after solving the equation ${e^{2x}} = 50$, we got the value of $x$ as $x = 1.956011$

Note: The key point to find the given equation is that when the equation consists of exponential terms, just take natural logarithm on both the sides of the equation as to solve for the value of x we need to remove the variable from the exponent by taking ln of the function. As Logarithmic functions are the inverses of exponential functions.