- Divide the equation by 4 to get \(\ln(9x) = 4\).
- Apply the exponential function \(e^y\) to both sides to remove the logarithm, yielding \(9x = e^4\).
- Solve for \(x\) by dividing by 9.
- Confirm the domain condition \(x > 0\) is satisfied.
- The final answer is \(\boxed{\frac{e^4}{9}}\).
Explanation
- Analyzing the Equation and Domain
We are given the logarithmic equation \(4 \ln(9x) = 16\). Our goal is to solve for \(x\). First, we must note the domain of the expression. The natural logarithm \(\ln(u)\) is defined only for \(u > 0\). Thus, for the term \(\ln(9x)\) to be defined, we must have \(9x > 0\), which simplifies to \(x > 0\). We will keep this condition in mind throughout our solution process.
- Isolating the Logarithmic Term
To solve for \(x\), we first isolate the logarithmic term \(\ln(9x)\). Dividing both sides of the original equation \(4 \ln(9x) = 16\) by \(4\), we obtain:
\[\ln(9x) = \frac{16}{4}\]
\[\ln(9x) = 4\]
- Converting to Exponential Form and Solving for x
The equation \(\ln(9x) = 4\) is in the logarithmic form \(\ln(u) = v\), which is equivalent to the exponential form \(u = e^v\). Applying this property to our equation:
\[9x = e^4\]
Now, to solve for \(x\), we divide both sides by \(9\):
\[x = \frac{e^4}{9}\]
- Verification and Conclusion
Finally, we check our solution against the domain constraint \(x > 0\). Since \(e^4\) is approximately \(54.598\), \(x = \frac{e^4}{9} \approx 6.066\), which is indeed greater than \(0\). Thus, the solution is valid. The final result is:
\[\boxed{\frac{e^4}{9}}\]
Examples
Logarithmic equations are commonly used to solve problems involving exponential growth or decay, such as calculating the time required for an investment to reach a certain value with continuous compounding interest or determining the age of an artifact using carbon dating. For instance, if a population grows according to \(P(t) = P_0 e^{kt}\), solving for the time \(t\) involves using logarithmic properties. Mastering these equations allows one to model and predict trends in finance, biology, and physics efficiently.