Algebra Examples

$2^{x + 1} + 2^{x} = 3^{y + 2} - 3^{y}$

Step 1

Factor out $2^{x}$ from the expression.

$2^{x} (2 + 1)$

Step 2

Add $2$ and $1$ .

$2^{x} \cdot 3 = 3^{y + 2} - 3^{y}$

Step 3

Move $3$ to the left of $2^{x}$ .

$3 \cdot 2^{x} = 3^{y + 2} - 3^{y}$

Step 4

Divide each term in $3 \cdot 2^{x} = 3^{y + 2} - 3^{y}$ by $3$ and simplify.

Tap for more steps...

Step 4.1

Divide each term in $3 \cdot 2^{x} = 3^{y + 2} - 3^{y}$ by $3$ .

$\frac{3 \cdot 2^{x}}{3} = \frac{3^{y + 2}}{3} + \frac{- 3^{y}}{3}$

Step 4.2

Simplify the left side.

Tap for more steps...

Step 4.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.2.1.1

Cancel the common factor.

$\frac{3 \cdot 2^{x}}{3} = \frac{3^{y + 2}}{3} + \frac{- 3^{y}}{3}$

Step 4.2.1.2

Divide $2^{x}$ by $1$ .

$2^{x} = \frac{3^{y + 2}}{3} + \frac{- 3^{y}}{3}$

Step 4.3

Simplify the right side.

Tap for more steps...

Step 4.3.1

Combine the numerators over the common denominator.

$2^{x} = \frac{3^{y + 2} - 3^{y}}{3}$

Step 5

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

$\ln (2^{x}) = \ln (\frac{3^{y + 2} - 3^{y}}{3})$

Step 6

Expand $\ln (2^{x})$ by moving $x$ outside the logarithm.

$x \ln (2) = \ln (\frac{3^{y + 2} - 3^{y}}{3})$

Step 7

Rewrite $\ln (\frac{3^{y + 2} - 3^{y}}{3})$ as $\ln (3^{y + 2} - 3^{y}) - \ln (3)$ .

$x \ln (2) = \ln (3^{y + 2} - 3^{y}) - \ln (3)$

Step 8

Simplify the right side.

Tap for more steps...

Step 8.1

Simplify $\ln (3^{y + 2} - 3^{y}) - \ln (3)$ .

Tap for more steps...

Step 8.1.1

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$x \ln (2) = \ln (\frac{3^{y + 2} - 3^{y}}{3})$

Step 8.1.2

Split the fraction $\frac{3^{y + 2} - 3^{y}}{3}$ into two fractions.

$x \ln (2) = \ln (\frac{3^{y + 2}}{3} + \frac{- 3^{y}}{3})$

Step 8.1.3

Simplify each term.

Tap for more steps...

Step 8.1.3.1

Cancel the common factor of $3^{y + 2}$ and $3$ .

Tap for more steps...

Step 8.1.3.1.1

Factor $3$ out of $3^{y + 2}$ .

$x \ln (2) = \ln (\frac{3 \cdot 3^{y + 1}}{3} + \frac{- 3^{y}}{3})$

Step 8.1.3.1.2

Cancel the common factors.

Tap for more steps...

Step 8.1.3.1.2.1

Factor $3$ out of $3$ .

$x \ln (2) = \ln (\frac{3 \cdot 3^{y + 1}}{3 (1)} + \frac{- 3^{y}}{3})$

Step 8.1.3.1.2.2

Cancel the common factor.

$x \ln (2) = \ln (\frac{3 \cdot 3^{y + 1}}{3 \cdot 1} + \frac{- 3^{y}}{3})$

Step 8.1.3.1.2.3

Rewrite the expression.

$x \ln (2) = \ln (\frac{3^{y + 1}}{1} + \frac{- 3^{y}}{3})$

Step 8.1.3.1.2.4

Divide $3^{y + 1}$ by $1$ .

$x \ln (2) = \ln (3^{y + 1} + \frac{- 3^{y}}{3})$

Step 8.1.3.2

Cancel the common factor of $3^{y}$ and $3$ .

Tap for more steps...

Step 8.1.3.2.1

Factor $3$ out of $- 3^{y}$ .

$x \ln (2) = \ln (3^{y + 1} + \frac{3 (- 3^{y - 1})}{3})$

Step 8.1.3.2.2

Cancel the common factors.

Tap for more steps...

Step 8.1.3.2.2.1

Factor $3$ out of $3$ .

$x \ln (2) = \ln (3^{y + 1} + \frac{3 (- 3^{y - 1})}{3 (1)})$

Step 8.1.3.2.2.2

Cancel the common factor.

$x \ln (2) = \ln (3^{y + 1} + \frac{3 (- 3^{y - 1})}{3 \cdot 1})$

Step 8.1.3.2.2.3

Rewrite the expression.

$x \ln (2) = \ln (3^{y + 1} + \frac{- 3^{y - 1}}{1})$

Step 8.1.3.2.2.4

Divide $- 3^{y - 1}$ by $1$ .

$x \ln (2) = \ln (3^{y + 1} - 3^{y - 1})$

Step 9

Divide each term in $x \ln (2) = \ln (3^{y + 1} - 3^{y - 1})$ by $\ln (2)$ and simplify.

Tap for more steps...

Step 9.1

Divide each term in $x \ln (2) = \ln (3^{y + 1} - 3^{y - 1})$ by $\ln (2)$ .

$\frac{x \ln (2)}{\ln (2)} = \frac{\ln (3^{y + 1} - 3^{y - 1})}{\ln (2)}$

Step 9.2

Simplify the left side.

Tap for more steps...

Step 9.2.1

Cancel the common factor of $\ln (2)$ .

Tap for more steps...

Step 9.2.1.1

Cancel the common factor.

$\frac{x \ln (2)}{\ln (2)} = \frac{\ln (3^{y + 1} - 3^{y - 1})}{\ln (2)}$

Step 9.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (3^{y + 1} - 3^{y - 1})}{\ln (2)}$