The given equation is:

$y = \ln(4x)$

Recall:

The natural logarithm $\ln$ is the logarithm to the base $e$, where $e \approx 2.718$. A logarithmic equation can be rewritten in exponential form as:

$\ln(a) = b \quad \implies \quad e^b = a$

Solution:

From $y = \ln(4x)$:

• The logarithmic base is $e$,
• The argument of the logarithm is $4x$,
• The result is $y$.

Rewriting this in exponential form:

$e^y = 4x$

Final Answer:

The equivalent exponential form is:

$e^y = 4x$

Let me know if you need further clarification or have additional questions!

Related Questions:

1. How do you convert $\log_b(x) = y$ to exponential form?
2. What is the relationship between $\ln(x)$ and the exponential function $e^x$?
3. How would you solve $e^y = 4x$ for $x$?
4. What is the value of $e$ and why is it important in mathematics?
5. How do you solve $\ln(x) = c$ for $x$?

Tip:

The natural logarithm $\ln(x)$ and the exponential function $e^x$ are inverses of each other, which makes converting between forms straightforward.