To solve the given logarithmic equation, we will start by applying inverse operations and the Properties of Equality to isolate the term with the logarithm.
2 ln 2x^2 = 1 ⇔ ln 2x^2 = 1/2
Now that we have isolated the term with the exponent, we will recall the definition of a logarithm.
log_b x=y ⇔ x= b^yThis tells us how we can rewrite the logarithm equivalent to y as an exponential equation. The argument x is equal to b raised to the power of y. The base of a natural logarithm is e, so ln x = log_e x. Let's rewrite the given equation in exponential form.
ln ( 2x^2)=1/2 ⇔ 2x^2= e^()darkviolet 12
Now, we can solve the equation for x.
The exact solutions are x=sqrt(e^(12)/2) and x = - sqrt(e^(12)/2). We can also write them in decimal form using a calculator.
sqrt(e^(12)/2) &≈ 0.908
- sqrt(e^(12)/2) &≈ - 0.908
Finally, we will check our answers by substituting 0.908 and - 0.908 for x in the given equation. Let's start with substituting 0.908.
