Algebra Examples

$7^{x + 1} = 3^{2 x + 2}$

Step 1

Take the log of both sides of the equation.

$\ln (7^{x + 1}) = \ln (3^{2 x + 2})$

Step 2

Expand $\ln (7^{x + 1})$ by moving $x + 1$ outside the logarithm.

$(x + 1) \ln (7) = \ln (3^{2 x + 2})$

Step 3

Expand $\ln (3^{2 x + 2})$ by moving $2 x + 2$ outside the logarithm.

$(x + 1) \ln (7) = (2 x + 2) \ln (3)$

Step 4

Solve the equation for $x$ .

Tap for more steps...

Step 4.1

Simplify $(x + 1) \ln (7)$ .

Tap for more steps...

Step 4.1.1

Rewrite.

$0 + 0 + (x + 1) \ln (7) = (2 x + 2) \ln (3)$

Step 4.1.2

Simplify by adding zeros.

$(x + 1) \ln (7) = (2 x + 2) \ln (3)$

Step 4.1.3

Apply the distributive property.

$x \ln (7) + 1 \ln (7) = (2 x + 2) \ln (3)$

Step 4.1.4

Multiply $\ln (7)$ by $1$ .

$x \ln (7) + \ln (7) = (2 x + 2) \ln (3)$

Step 4.2

Apply the distributive property.

$x \ln (7) + \ln (7) = 2 x \ln (3) + 2 \ln (3)$

Step 4.3

Subtract $2 x \ln (3)$ from both sides of the equation.

$x \ln (7) + \ln (7) - 2 x \ln (3) = 2 \ln (3)$

Step 4.4

Subtract $\ln (7)$ from both sides of the equation.

$x \ln (7) - 2 x \ln (3) = 2 \ln (3) - \ln (7)$

Step 4.5

Factor $x$ out of $x \ln (7) - 2 x \ln (3)$ .

Tap for more steps...

Step 4.5.1

Factor $x$ out of $x \ln (7)$ .

$x (\ln (7)) - 2 x \ln (3) = 2 \ln (3) - \ln (7)$

Step 4.5.2

Factor $x$ out of $- 2 x \ln (3)$ .

$x (\ln (7)) + x (- 2 \ln (3)) = 2 \ln (3) - \ln (7)$

Step 4.5.3

Factor $x$ out of $x (\ln (7)) + x (- 2 \ln (3))$ .

$x (\ln (7) - 2 \ln (3)) = 2 \ln (3) - \ln (7)$

Step 4.6

Divide each term in $x (\ln (7) - 2 \ln (3)) = 2 \ln (3) - \ln (7)$ by $\ln (7) - 2 \ln (3)$ and simplify.

Tap for more steps...

Step 4.6.1

Divide each term in $x (\ln (7) - 2 \ln (3)) = 2 \ln (3) - \ln (7)$ by $\ln (7) - 2 \ln (3)$ .

$\frac{x (\ln (7) - 2 \ln (3))}{\ln (7) - 2 \ln (3)} = \frac{2 \ln (3)}{\ln (7) - 2 \ln (3)} + \frac{- \ln (7)}{\ln (7) - 2 \ln (3)}$

Step 4.6.2

Simplify the left side.

Tap for more steps...

Step 4.6.2.1

Cancel the common factor of $\ln (7) - 2 \ln (3)$ .

Tap for more steps...

Step 4.6.2.1.1

Cancel the common factor.

$\frac{x (\ln (7) - 2 \ln (3))}{\ln (7) - 2 \ln (3)} = \frac{2 \ln (3)}{\ln (7) - 2 \ln (3)} + \frac{- \ln (7)}{\ln (7) - 2 \ln (3)}$

Step 4.6.2.1.2

Divide $x$ by $1$ .

$x = \frac{2 \ln (3)}{\ln (7) - 2 \ln (3)} + \frac{- \ln (7)}{\ln (7) - 2 \ln (3)}$

Step 4.6.3

Simplify the right side.

Tap for more steps...

Step 4.6.3.1

Combine the numerators over the common denominator.

$x = \frac{2 \ln (3) - \ln (7)}{\ln (7) - 2 \ln (3)}$

Step 4.6.3.2

Cancel the common factor of $2 \ln (3) - \ln (7)$ and $\ln (7) - 2 \ln (3)$ .

Tap for more steps...

Step 4.6.3.2.1

Factor $- 1$ out of $2 \ln (3)$ .

$x = \frac{- (- 2 \ln (3)) - \ln (7)}{\ln (7) - 2 \ln (3)}$

Step 4.6.3.2.2

Factor $- 1$ out of $- \ln (7)$ .

$x = \frac{- (- 2 \ln (3)) - \ln (7)}{\ln (7) - 2 \ln (3)}$

Step 4.6.3.2.3

Factor $- 1$ out of $- (- 2 \ln (3)) - \ln (7)$ .

$x = \frac{- (- 2 \ln (3) + \ln (7))}{\ln (7) - 2 \ln (3)}$

Step 4.6.3.2.4

Rewrite $- (- 2 \ln (3) + \ln (7))$ as $- 1 (- 2 \ln (3) + \ln (7))$ .

$x = \frac{- 1 (- 2 \ln (3) + \ln (7))}{\ln (7) - 2 \ln (3)}$

Step 4.6.3.2.5

Reorder terms.

$x = \frac{- 1 (- 2 \ln (3) + \ln (7))}{- 2 \ln (3) + \ln (7)}$

Step 4.6.3.2.6

Cancel the common factor.

$x = \frac{- 1 (- 2 \ln (3) + \ln (7))}{- 2 \ln (3) + \ln (7)}$

Step 4.6.3.2.7

Divide $- 1$ by $1$ .

$x = - 1$