Solve the following logarithmic equation.

Question

4 plus ln left parenthesis 8 x right parenthesis equals 25 minus 2 ln left parenthesis x right parenthesis
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Part 1
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​(Simplify your answer. Type an exact​ answer, using e as​ needed.)

Answer 1

To solve the equation \( 4 + \ln(8x) = 25 - 2\ln(x) \), we will first move all logarithmic terms to one side and the constant terms to the other side.

1. Rearranging the equation: \[ \ln(8x) + 2\ln(x) = 25 - 4 \] \[ \ln(8x) + 2\ln(x) = 21 \]

2. Using the properties of logarithms, we can combine the logarithmic terms on the left side:

   • Recall \( \ln(a) + \ln(b) = \ln(ab) \),
   • Also, \( 2\ln(x) = \ln(x^2) \).

   Therefore: \[ \ln(8x) + \ln(x^2) = \ln(8x \cdot x^2) = \ln(8x^3) \]

   So we have: \[ \ln(8x^3) = 21 \]

3. Exponentiating both sides to eliminate the logarithm: \[ 8x^3 = e^{21} \]

4. Solving for \( x^3 \): \[ x^3 = \frac{e^{21}}{8} \]

5. Finally, to solve for \( x \): \[ x = \left( \frac{e^{21}}{8} \right)^{\frac{1}{3}} = \frac{e^{21/3}}{2} = \frac{e^{7}}{2} \]

Thus, the solution for \( x \) is: \[ \boxed{\frac{e^7}{2}} \]