Question

solve: 4 ln x = 2 ln 25
○ x = 5
○ x = 25
× x = 625
retry

Explanation:

Step1: Simplify using logarithm power rule

Using the power rule of logarithms \(a\ln b=\ln(b^a)\), we can rewrite the given equation \(4\ln x = 2\ln25\) as \(\ln(x^4)=\ln(25^2)\).

Step2: Use one - to - one property of logarithms

If \(\ln a=\ln b\), then \(a = b\) (since the natural logarithm function \(y = \ln x\) is one - to - one). So we have the equation \(x^4=25^2\).
We know that \(25^2 = 625\), so \(x^4=625\). But we can also simplify the original equation in another way. Let's divide both sides of the original equation \(4\ln x=2\ln25\) by 2 first. We get \(2\ln x=\ln25\). Then, using the power rule again, \(\ln(x^2)=\ln25\).
By the one - to - one property of logarithms, \(x^2 = 25\) (we consider the positive root since the domain of the natural logarithm function \(y=\ln x\) requires \(x>0\)). Then, taking the square root of both sides, \(x = 5\) (we discard the negative root because \(x>0\) for \(\ln x\) to be defined).

Answer:

\(x = 5\) (the option \(x = 5\))