solve: 4 ln x = 2 ln 25\nx = 5\nx = 25\nx = 625\ndone

Explanation:

Step1: Use logarithm property

Using the power - rule of logarithms (n\ln a=\ln(a^{n})), we can rewrite the left - hand side as (\ln(x^{4})) and the right - hand side as (\ln(25^{2})). So the equation (4\ln x = 2\ln25) becomes (\ln(x^{4})=\ln(25^{2})).

Step2: Eliminate the logarithms

Since the natural logarithm function (y = \ln u) is one - to - one, if (\ln(x^{4})=\ln(25^{2})), then (x^{4}=25^{2}). Calculate (25^{2}=625), so (x^{4}=625).

Step3: Solve for (x)

Take the fourth - root of both sides. Since (x>0) (because (\ln x) is only defined for (x > 0)), (x=\sqrt[4]{625}). And (\sqrt[4]{625}=5) because (5\times5\times5\times5 = 625).

Answer:

(x = 5)