Precalculus Examples

${(2^{x + 5} \cdot 2^{- 9})}^{2} = 2$

Step 1

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

$\ln ({(2^{x + 5} \cdot 2^{- 9})}^{2}) = \ln (2)$

Step 2

Expand the left side.

Tap for more steps...

Step 2.1

Expand $\ln ({(2^{x + 5} \cdot 2^{- 9})}^{2})$ by moving $2$ outside the logarithm.

$2 \ln (2^{x + 5} \cdot 2^{- 9}) = \ln (2)$

Step 2.2

Multiply $2^{x + 5}$ by $2^{- 9}$ by adding the exponents.

Tap for more steps...

Step 2.2.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 \ln (2^{x + 5 - 9}) = \ln (2)$

Step 2.2.2

Subtract $9$ from $5$ .

$2 \ln (2^{x - 4}) = \ln (2)$

Step 2.3

Expand $\ln (2^{x - 4})$ by moving $x - 4$ outside the logarithm.

$2 ((x - 4) \ln (2)) = \ln (2)$

Step 3

Simplify the left side.

Tap for more steps...

Step 3.1

Simplify $2 ((x - 4) \ln (2))$ .

Tap for more steps...

Step 3.1.1

Apply the distributive property.

$2 (x \ln (2) - 4 \ln (2)) = \ln (2)$

Step 3.1.2

Apply the distributive property.

$2 (x \ln (2)) + 2 (- 4 \ln (2)) = \ln (2)$

Step 3.1.3

Multiply $- 4$ by $2$ .

$2 x \ln (2) - 8 \ln (2) = \ln (2)$

Step 4

Move all the terms containing a logarithm to the left side of the equation.

$2 x \ln (2) - 8 \ln (2) - \ln (2) = 0$

Step 5

Subtract $\ln (2)$ from $- 8 \ln (2)$ .

$2 x \ln (2) - 9 \ln (2) = 0$

Step 6

Add $9 \ln (2)$ to both sides of the equation.

$2 x \ln (2) = 9 \ln (2)$

Step 7

Divide each term in $2 x \ln (2) = 9 \ln (2)$ by $2 \ln (2)$ and simplify.

Tap for more steps...

Step 7.1

Divide each term in $2 x \ln (2) = 9 \ln (2)$ by $2 \ln (2)$ .

$\frac{2 x \ln (2)}{2 \ln (2)} = \frac{9 \ln (2)}{2 \ln (2)}$

Step 7.2

Simplify the left side.

Tap for more steps...

Step 7.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 7.2.1.1

Cancel the common factor.

$\frac{2 x \ln (2)}{2 \ln (2)} = \frac{9 \ln (2)}{2 \ln (2)}$

Step 7.2.1.2

Rewrite the expression.

$\frac{x \ln (2)}{\ln (2)} = \frac{9 \ln (2)}{2 \ln (2)}$

Step 7.2.2

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

Tap for more steps...

Step 7.2.2.1

Cancel the common factor.

$\frac{x \ln (2)}{\ln (2)} = \frac{9 \ln (2)}{2 \ln (2)}$

Step 7.2.2.2

Divide $x$ by $1$ .

$x = \frac{9 \ln (2)}{2 \ln (2)}$

Step 7.3

Simplify the right side.

Tap for more steps...

Step 7.3.1

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

Tap for more steps...

Step 7.3.1.1

Cancel the common factor.

$x = \frac{9 \ln (2)}{2 \ln (2)}$

Step 7.3.1.2

Rewrite the expression.

$x = \frac{9}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \frac{9}{2}$

Decimal Form:

$x = 4.5$

Mixed Number Form:

$x = 4 \frac{1}{2}$