Question

ng logarithmic equations
solve: $4\ln x = 2\ln 25$
$x = 625$
$x = 25$
$x = 5$

Explanation:

Step1: Simplify using logarithm power rule

Using the power rule of logarithms \( n\ln a=\ln(a^n) \), 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 from \( \ln(x^4)=\ln(25^2) \), we have:
\( x^4=25^2 \)
We know that \( 25^2=(5^2)^2 = 5^4 \), so \( x^4 = 5^4 \)

Step3: Solve for x

Taking the fourth root of both sides, and considering the domain of the natural logarithm ( \( x>0 \) ), we get \( x = 5 \) (since for real numbers, if \( a^n=b^n \) and \( n \) is even, \( a=\pm b \), but \( x>0 \), so we take the positive root).

Answer:

\( x = 5 \)