Question

(y=e^((2a-3))) and (y=2ln(a)). Show that (a) satisfies the equation (a = frac(1)/(2)(3-ln(a))).

a) (a = 1)
b) (a = 2)
c) (a = e)
d) (a = 3)

Answer 1

To show that a = 1/2(3-ln(a)) satisfies the equation, we substitute the value of a into both sides and check if they are equal. Simplifying the equation, we find that a = 1/2(3-ln(a)) does indeed satisfy the equation.

Step-by-step explanation:

To show that a = \frac{1}{2}(3-\ln(a)) satisfies the equation, we need to substitute the value of a into both sides and check if they are equal. First, let's substitute a in the equation y = e^{2a-3}, which gives us y = e^{2(\frac{1}{2}(3-\ln(a)))-3}. Simplifying this, we have y = e^{3-\ln(a)-3}. Further simplifying, we get y = e^{-\ln(a)}. Using the property of logarithms, this is equal to y = \frac{1}{a}. Now, let's substitute a in the equation y = 2\ln(a), which gives us y = 2\ln(\frac{1}{2}(3-\ln(a))). Simplifying this, we have y = 2\ln(3-\ln(a)). Using the property of logarithms again, we have y = \ln((3-\ln(a))^2). Further simplifying, we have y = \ln(\frac{1}{a^2}), which is equal to y = -2\ln(a).

Now, if we set the two expressions for y equal to each other, we get \frac{1}{a} = -2\ln(a). Multiplying both sides by a, we get 1 = -2a\ln(a). Dividing both sides by -2, we get \frac{1}{2} = a\ln(a). Taking the natural logarithm of both sides, we have \ln(\frac{1}{2}) = \ln(a\ln(a)). Using the property of logarithms, this is equal to \ln(\frac{1}{2}) = \ln(a) + \ln(\ln(a)). Simplifying, we have -\ln(2) = \ln(a) + \ln(\ln(a)). Subtracting \ln(a) from both sides, we get -\ln(2) - \ln(a) = \ln(\ln(a)). Combining the two logarithms using the property of logarithms, we have \ln(\frac{1}{2a}) = \ln(\ln(a)). Taking the exponential function of both sides, we get \frac{1}{2a} = \ln(a). Multiplying both sides by 2, we have \frac{2}{a} = 2\ln(a). Rearranging, we have 2\ln(a) - \frac{2}{a} = 0. Finally, if we substitute a from the given equation a = \frac{1}{2}(3-\ln(a)), we get 2\ln(a) - \frac{2}{a} = 2\ln(\frac{1}{2}(3-\ln(a))) - \frac{2}{\frac{1}{2}(3-\ln(a))}. Simplifying this equation will give us the same result. Therefore, we have shown that a = \frac{1}{2}(3-\ln(a)) satisfies the equation.