Solve for x 4*8^x=11.48

$4 \cdot 8^{x} = 11.48$

Step 1

Simplify.

Rewrite $4$ as $2^{2}$ .

$2^{2} \cdot 8^{x} = 11.48$

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

$2^{2} \cdot {(2^{3})}^{x} = 11.48$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$2^{2} \cdot 2^{3 x} = 11.48$

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

$2^{2 + 3 x} = 11.48$

Step 2

Divide each term in $2^{2 + 3 x} = 11.48$ by $4$ and simplify.

Divide each term in $2^{2 + 3 x} = 11.48$ by $4$ .

$\frac{2^{2 + 3 x}}{4} = \frac{11.48}{4}$

Simplify the right side.

Divide $11.48$ by $4$ .

$\frac{2^{2 + 3 x}}{4} = 2.87$

Step 3

Multiply both sides by $4$ .

$\frac{2^{2 + 3 x}}{4} \cdot 4 = 2.87 \cdot 4$

Step 4

Simplify.

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{2^{2 + 3 x}}{4} \cdot 4 = 2.87 \cdot 4$

Rewrite the expression.

$2^{2 + 3 x} = 2.87 \cdot 4$

Simplify the right side.

Multiply $2.87$ by $4$ .

$2^{2 + 3 x} = 11.48$

Step 5

Solve for $x$ .

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

$\ln (2^{2 + 3 x}) = \ln (11.48)$

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

$(2 + 3 x) \ln (2) = \ln (11.48)$

Simplify the left side.

Apply the distributive property.

$2 \ln (2) + 3 x \ln (2) = \ln (11.48)$

Reorder $2 \ln (2)$ and $3 x \ln (2)$ .

$3 x \ln (2) + 2 \ln (2) = \ln (11.48)$

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

$3 x \ln (2) + 2 \ln (2) - \ln (11.48) = 0$

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

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

$3 x \ln (2) - \ln (11.48) = - 2 \ln (2)$

Add $\ln (11.48)$ to both sides of the equation.

$3 x \ln (2) = - 2 \ln (2) + \ln (11.48)$

Divide each term in $3 x \ln (2) = - 2 \ln (2) + \ln (11.48)$ by $3 \ln (2)$ and simplify.

Divide each term in $3 x \ln (2) = - 2 \ln (2) + \ln (11.48)$ by $3 \ln (2)$ .

$\frac{3 x \ln (2)}{3 \ln (2)} = \frac{- 2 \ln (2)}{3 \ln (2)} + \frac{\ln (11.48)}{3 \ln (2)}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x \ln (2)}{3 \ln (2)} = \frac{- 2 \ln (2)}{3 \ln (2)} + \frac{\ln (11.48)}{3 \ln (2)}$

Rewrite the expression.

$\frac{x \ln (2)}{\ln (2)} = \frac{- 2 \ln (2)}{3 \ln (2)} + \frac{\ln (11.48)}{3 \ln (2)}$

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

Cancel the common factor.

$\frac{x \ln (2)}{\ln (2)} = \frac{- 2 \ln (2)}{3 \ln (2)} + \frac{\ln (11.48)}{3 \ln (2)}$

Divide $x$ by $1$ .

$x = \frac{- 2 \ln (2)}{3 \ln (2)} + \frac{\ln (11.48)}{3 \ln (2)}$

Simplify the right side.

Simplify each term.

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

Cancel the common factor.

$x = \frac{- 2 \ln (2)}{3 \ln (2)} + \frac{\ln (11.48)}{3 \ln (2)}$

Rewrite the expression.

$x = \frac{- 2}{3} + \frac{\ln (11.48)}{3 \ln (2)}$

Move the negative in front of the fraction.

$x = - \frac{2}{3} + \frac{\ln (11.48)}{3 \ln (2)}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{2}{3} + \frac{\ln (11.48)}{3 \ln (2)}$

Decimal Form:

$x = 0.50701691 \dots$

Solve for x 4*8^x=11.48

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