Solve for x ((4^6)/(4^x))^3=4^6

${(\frac{4^{6}}{4^{x}})}^{3} = 4^{6}$

Step 1

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

$\ln ({(\frac{4^{6}}{4^{x}})}^{3}) = \ln (4^{6})$

Step 2

Expand the left side.

Expand $\ln ({(\frac{4^{6}}{4^{x}})}^{3})$ by moving $3$ outside the logarithm.

$3 \ln (\frac{4^{6}}{4^{x}}) = \ln (4^{6})$

Reduce the expression $\frac{4^{6}}{4^{x}}$ by cancelling the common factors.

Factor $4^{x}$ out of $4^{6}$ .

$3 \ln (\frac{4^{x} \cdot 4^{6 - x}}{4^{x}}) = \ln (4^{6})$

Multiply by $1$ .

$3 \ln (\frac{4^{x} \cdot 4^{6 - x}}{4^{x} \cdot 1}) = \ln (4^{6})$

Cancel the common factor.

$3 \ln (\frac{4^{x} \cdot 4^{6 - x}}{4^{x} \cdot 1}) = \ln (4^{6})$

Rewrite the expression.

$3 \ln (\frac{4^{6 - x}}{1}) = \ln (4^{6})$

Divide $4^{6 - x}$ by $1$ .

$3 \ln (4^{6 - x}) = \ln (4^{6})$

Expand $\ln (4^{6 - x})$ by moving $6 - x$ outside the logarithm.

$3 ((6 - x) \ln (4)) = \ln (4^{6})$

Step 3

Simplify the left side.

Simplify $3 ((6 - x) \ln (4))$ .

Apply the distributive property.

$3 (6 \ln (4) - x \ln (4)) = \ln (4^{6})$

Apply the distributive property.

$3 (6 \ln (4)) + 3 (- x \ln (4)) = \ln (4^{6})$

Multiply.

Multiply $6$ by $3$ .

$18 \ln (4) + 3 (- x \ln (4)) = \ln (4^{6})$

Multiply $- 1$ by $3$ .

$18 \ln (4) - 3 x \ln (4) = \ln (4^{6})$

Step 4

Simplify the right side.

Raise $4$ to the power of $6$ .

$18 \ln (4) - 3 x \ln (4) = \ln (4096)$

Step 5

Reorder $18 \ln (4)$ and $- 3 x \ln (4)$ .

$- 3 x \ln (4) + 18 \ln (4) = \ln (4096)$

Step 6

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

$- 3 x \ln (4) + 18 \ln (4) - \ln (4096) = 0$

Step 7

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

Subtract $18 \ln (4)$ from both sides of the equation.

$- 3 x \ln (4) - \ln (4096) = - 18 \ln (4)$

Add $\ln (4096)$ to both sides of the equation.

$- 3 x \ln (4) = - 18 \ln (4) + \ln (4096)$

Step 8

Divide each term in $- 3 x \ln (4) = - 18 \ln (4) + \ln (4096)$ by $- 3 \ln (4)$ and simplify.

Divide each term in $- 3 x \ln (4) = - 18 \ln (4) + \ln (4096)$ by $- 3 \ln (4)$ .

$\frac{- 3 x \ln (4)}{- 3 \ln (4)} = \frac{- 18 \ln (4)}{- 3 \ln (4)} + \frac{\ln (4096)}{- 3 \ln (4)}$

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

$\frac{- 3 x \ln (4)}{- 3 \ln (4)} = \frac{- 18 \ln (4)}{- 3 \ln (4)} + \frac{\ln (4096)}{- 3 \ln (4)}$

Rewrite the expression.

$\frac{x \ln (4)}{\ln (4)} = \frac{- 18 \ln (4)}{- 3 \ln (4)} + \frac{\ln (4096)}{- 3 \ln (4)}$

Cancel the common factor of $\ln (4)$ .

Cancel the common factor.

$\frac{x \ln (4)}{\ln (4)} = \frac{- 18 \ln (4)}{- 3 \ln (4)} + \frac{\ln (4096)}{- 3 \ln (4)}$

Divide $x$ by $1$ .

$x = \frac{- 18 \ln (4)}{- 3 \ln (4)} + \frac{\ln (4096)}{- 3 \ln (4)}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $- 18$ and $- 3$ .

Factor $- 3$ out of $- 18 \ln (4)$ .

$x = \frac{- 3 (6 \ln (4))}{- 3 \ln (4)} + \frac{\ln (4096)}{- 3 \ln (4)}$

Cancel the common factors.

Factor $- 3$ out of $- 3 \ln (4)$ .

$x = \frac{- 3 (6 \ln (4))}{- 3 \ln (4)} + \frac{\ln (4096)}{- 3 \ln (4)}$

Cancel the common factor.

$x = \frac{- 3 (6 \ln (4))}{- 3 \ln (4)} + \frac{\ln (4096)}{- 3 \ln (4)}$

Rewrite the expression.

$x = \frac{6 \ln (4)}{\ln (4)} + \frac{\ln (4096)}{- 3 \ln (4)}$

Cancel the common factor of $\ln (4)$ .

Cancel the common factor.

$x = \frac{6 \ln (4)}{\ln (4)} + \frac{\ln (4096)}{- 3 \ln (4)}$

Divide $6$ by $1$ .

$x = 6 + \frac{\ln (4096)}{- 3 \ln (4)}$

Move the negative in front of the fraction.

$x = 6 - \frac{\ln (4096)}{3 \ln (4)}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = 6 - \frac{\ln (4096)}{3 \ln (4)}$

Decimal Form:

$x = 4$

Solve for x ((4^6)/(4^x))^3=4^6

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