Question\ln |y| = \ln |x| + c \\ \text{Solve for } y

Studdy Solution

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:
$|y| = |x| \cdot e^c$

STEP 5

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:
$y = \pm |x| \cdot e^c$

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms