Question

solve: \\(4 \ln x = 2 \ln 25\\)

\\(x = 5\\)

\\(x = 25\\)

\\(x = 625\\)

Explanation:

⚡ Using what you learned: Logarithms and Exponential Equations

Step 1: Isolate the logarithm term

Divide both sides of the equation by \(4\):

$$ \ln x = \frac{2 \ln 25}{4} $$

$$ \ln x = \frac{1}{2} \ln 25 $$

Step 2: Apply the power property of logarithms

Use the property \(b \ln a = \ln(a^b)\):

$$ \ln x = \ln(25^{1/2}) $$

$$ \ln x = \ln 5 $$

Step 3: Solve for x

Since the natural logarithms are equal, their arguments must be equal:

$$ x = 5 $$

Answer:

\(x = 5\)