what is the true solution to $ln 20+ln 5 = 2ln x$?\n$x = 5$\n$x = 10$\n$x = 50$\n$x = 100$

Explanation:

Step1: Apply logarithm addition rule

According to the rule $\ln a+\ln b=\ln(ab)$, we have $\ln(20\times5)=\ln100$. And $2\ln x = \ln(x^{2})$ by the power - rule of logarithms $n\ln a=\ln(a^{n})$. So the equation becomes $\ln100=\ln(x^{2})$.

Step2: Solve for x

Since the natural - logarithm function $y = \ln u$ is one - to - one, if $\ln a=\ln b$, then $a = b$. So $x^{2}=100$. Taking the square root of both sides, we get $x=\pm10$. But since the domain of the natural logarithm function $\ln x$ is $x>0$, we discard $x = - 10$. So $x = 10$.

Answer:

$x = 10$