Calculus Examples

$e^{a x} = C e^{b x}$

Step 1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{a x}) = \ln (C e^{b x})$

Step 2

Expand the left side.

Tap for more steps...

Step 2.1

Expand $\ln (e^{a x})$ by moving $a x$ outside the logarithm.

$a x \ln (e) = \ln (C e^{b x})$

Step 2.2

The natural logarithm of $e$ is $1$ .

$a x \cdot 1 = \ln (C e^{b x})$

Step 2.3

Multiply $a$ by $1$ .

$a x = \ln (C e^{b x})$

Step 3

Expand the right side.

Tap for more steps...

Step 3.1

Rewrite $\ln (C e^{b x})$ as $\ln (C) + \ln (e^{b x})$ .

$a x = \ln (C) + \ln (e^{b x})$

Step 3.2

Expand $\ln (e^{b x})$ by moving $b x$ outside the logarithm.

$a x = \ln (C) + b x \ln (e)$

Step 3.3

The natural logarithm of $e$ is $1$ .

$a x = \ln (C) + b x \cdot 1$

Step 3.4

Multiply $b$ by $1$ .

$a x = \ln (C) + b x$

Step 4

Subtract $b x$ from both sides of the equation.

$a x - b x = \ln (C)$

Step 5

Factor $x$ out of $a x - b x$ .

Tap for more steps...

Step 5.1

Factor $x$ out of $a x$ .

$x a - b x = \ln (C)$

Step 5.2

Factor $x$ out of $- b x$ .

$x a + x (- b) = \ln (C)$

Step 5.3

Factor $x$ out of $x a + x (- b)$ .

$x (a - b) = \ln (C)$

Step 6

Divide each term in $x (a - b) = \ln (C)$ by $a - b$ and simplify.

Tap for more steps...

Step 6.1

Divide each term in $x (a - b) = \ln (C)$ by $a - b$ .

$\frac{x (a - b)}{a - b} = \frac{\ln (C)}{a - b}$

Step 6.2

Simplify the left side.

Tap for more steps...

Step 6.2.1

Cancel the common factor of $a - b$ .

Tap for more steps...

Step 6.2.1.1

Cancel the common factor.

$\frac{x (a - b)}{a - b} = \frac{\ln (C)}{a - b}$

Step 6.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (C)}{a - b}$