Solve for x 8(2^(x+3))=48

$8 (2^{x + 3}) = 48$

Step 1

Simplify.

Rewrite $8$ as $2^{3}$ .

$2^{3} \cdot 2^{x + 3} = 48$

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

$2^{3 + x + 3} = 48$

Add $3$ and $3$ .

$2^{x + 6} = 48$

Step 2

Divide each term in $2^{x + 6} = 48$ by $8$ and simplify.

Divide each term in $2^{x + 6} = 48$ by $8$ .

$\frac{2^{x + 6}}{8} = \frac{48}{8}$

Simplify the right side.

Divide $48$ by $8$ .

$\frac{2^{x + 6}}{8} = 6$

Step 3

Multiply both sides of the equation by $8$ .

$8 \frac{2^{x + 6}}{8} = 8 \cdot 6$

Step 4

Simplify both sides of the equation.

Simplify the left side.

Cancel the common factor of $8$ .

Cancel the common factor.

$8 \frac{2^{x + 6}}{8} = 8 \cdot 6$

Rewrite the expression.

$2^{x + 6} = 8 \cdot 6$

Simplify the right side.

Multiply $8$ by $6$ .

$2^{x + 6} = 48$

Step 5

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

$\ln (2^{x + 6}) = \ln (48)$

Step 6

Expand $\ln (2^{x + 6})$ by moving $x + 6$ outside the logarithm.

$(x + 6) \ln (2) = \ln (48)$

Step 7

Simplify the left side.

Apply the distributive property.

$x \ln (2) + 6 \ln (2) = \ln (48)$

Step 8

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

$x \ln (2) + 6 \ln (2) - \ln (48) = 0$

Step 9

Move all terms not containing $x$ to the right side of the equation.

Subtract $6 \ln (2)$ from both sides of the equation.

$x \ln (2) - \ln (48) = - 6 \ln (2)$

Add $\ln (48)$ to both sides of the equation.

$x \ln (2) = - 6 \ln (2) + \ln (48)$

Step 10

Divide each term in $x \ln (2) = - 6 \ln (2) + \ln (48)$ by $\ln (2)$ and simplify.

Divide each term in $x \ln (2) = - 6 \ln (2) + \ln (48)$ by $\ln (2)$ .

$\frac{x \ln (2)}{\ln (2)} = \frac{- 6 \ln (2)}{\ln (2)} + \frac{\ln (48)}{\ln (2)}$

Simplify the left side.

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

Cancel the common factor.

$\frac{x \ln (2)}{\ln (2)} = \frac{- 6 \ln (2)}{\ln (2)} + \frac{\ln (48)}{\ln (2)}$

Divide $x$ by $1$ .

$x = \frac{- 6 \ln (2)}{\ln (2)} + \frac{\ln (48)}{\ln (2)}$

Simplify the right side.

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

Cancel the common factor.

$x = \frac{- 6 \ln (2)}{\ln (2)} + \frac{\ln (48)}{\ln (2)}$

Divide $- 6$ by $1$ .

$x = - 6 + \frac{\ln (48)}{\ln (2)}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$x = - 6 + \frac{\ln (48)}{\ln (2)}$

Decimal Form:

$x = - 0.41503749 \dots$

Solve for x 8(2^(x+3))=48

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