Problem

Solve the logarithmic equation.
\[
3+\ln (8 x)=21-2 \ln x
\]

Rewrite the given equation without logarithms. Do not solve for $x$.
(Type an equation. Type an exact answer. Do not simplify.)

Answer

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Answer

The given logarithmic equation can be rewritten without logarithms as: \(\boxed{8 x^3 = e^{18}}\)

Steps

Step 1 :Isolate the logarithmic terms on one side of the equation and the non-logarithmic terms on the other side: \(\ln (8 x) + 2 \ln x = 21 - 3\)

Step 2 :Use the property of logarithms that says the logarithm of a product is the sum of the logarithms: \(\ln (8 x^3) = 18\)

Step 3 :Rewrite the logarithmic term as an exponential term: \(8 x^3 = e^{18}\)

Step 4 :The given logarithmic equation can be rewritten without logarithms as: \(\boxed{8 x^3 = e^{18}}\)

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