Math Snap

STEP 1

1. The equation involves natural logarithms.
2. We assume $c$ is a constant.
3. We are solving for $y$ in terms of $x$ and $c$.

STEP 2

1. Use properties of logarithms to simplify the equation.
2. Solve for the absolute value of $y$.
3. Consider the absolute value to solve for $y$.

STEP 3

Use the property of logarithms that states $\ln a + \ln b = \ln(ab)$. Here, we can rewrite the right side of the equation:
$$\ln |y| = \ln |x| + c$$ Rewriting $c$ as $\ln e^c$, we have:
$$\ln |y| = \ln (|x| \cdot e^c)$$

STEP 4

Since the natural logarithm function is one-to-one, if $\ln a = \ln b$, then $a = b$. Therefore, we equate the arguments of the logarithms:
$$|y| = |x| \cdot e^c$$

Consider the absolute value to solve for $y$. Since $|y| = |x| \cdot e^c$, we have two cases for $y$:
1. $y = |x| \cdot e^c$
2. $y = -|x| \cdot e^c$
Thus, the solution for $y$ is:
$$y = \pm |x| \cdot e^c$$ The solution for $y$ is:
$$y = \pm |x| \cdot e^c$$