Solve for x 2 log base 4 of 64=3 log base 4 of x

$2 \log_{4} (64) = 3 \log_{4} (x)$

Rewrite the equation as $3 \log_{4} (x) = 2 \log_{4} (64)$ .

$3 \log_{4} (x) = 2 \log_{4} (64)$

Simplify.

Simplify $3 \log_{4} (x)$ by moving $3$ inside the logarithm.

$\log_{4} (x^{3}) = 2 \log_{4} (64)$

Simplify $2 \log_{4} (64)$ by moving $2$ inside the logarithm.

$\log_{4} (x^{3}) = \log_{4} (64^{2})$

For the equation to be equal, the argument of the logarithms on both sides of the equation must be equal.

$x^{3} = 64^{2}$

Solve for $x$ .

Subtract $64^{2}$ from both sides of the equation.

$x^{3} - 64^{2} = 0$

Simplify each term.

Raise $64$ to the power of $2$ .

$x^{3} - 1 \cdot 4096 = 0$

Multiply $- 1$ by $4096$ .

$x^{3} - 4096 = 0$

Factor the left side of the equation.

Rewrite $4096$ as $16^{3}$ .

$x^{3} - 16^{3} = 0$

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = x$ and $b = 16$ .

$(x - 16) (x^{2} + x \cdot 16 + 16^{2}) = 0$

Simplify.

Move $16$ to the left of $x$ .

$(x - 16) (x^{2} + 16 x + 16^{2}) = 0$

Raise $16$ to the power of $2$ .

$(x - 16) (x^{2} + 16 x + 256) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 16 = 0$

$x^{2} + 16 x + 256 = 0$

Set $x - 16$ equal to $0$ and solve for $x$ .

Set $x - 16$ equal to $0$ .

$x - 16 = 0$

Add $16$ to both sides of the equation.

$x = 16$

Set $x^{2} + 16 x + 256$ equal to $0$ and solve for $x$ .

Set $x^{2} + 16 x + 256$ equal to $0$ .

$x^{2} + 16 x + 256 = 0$

Solve $x^{2} + 16 x + 256 = 0$ for $x$ .

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = 1$ , $b = 16$ , and $c = 256$ into the quadratic formula and solve for $x$ .

$\frac{- 16 \pm \sqrt{16^{2} - 4 \cdot (1 \cdot 256)}}{2 \cdot 1}$

Simplify.

Simplify the numerator.

Raise $16$ to the power of $2$ .

$x = \frac{- 16 \pm \sqrt{256 - 4 \cdot 1 \cdot 256}}{2 \cdot 1}$

Multiply $- 4$ by $1$ .

$x = \frac{- 16 \pm \sqrt{256 - 4 \cdot 256}}{2 \cdot 1}$

Multiply $- 4$ by $256$ .

$x = \frac{- 16 \pm \sqrt{256 - 1024}}{2 \cdot 1}$

Subtract $1024$ from $256$ .

$x = \frac{- 16 \pm \sqrt{- 768}}{2 \cdot 1}$

Rewrite $- 768$ as $- 1 (768)$ .

$x = \frac{- 16 \pm \sqrt{- 1 \cdot 768}}{2 \cdot 1}$

Rewrite $\sqrt{- 1 (768)}$ as $\sqrt{- 1} \cdot \sqrt{768}$ .

$x = \frac{- 16 \pm \sqrt{- 1} \cdot \sqrt{768}}{2 \cdot 1}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 16 \pm i \cdot \sqrt{768}}{2 \cdot 1}$

Rewrite $768$ as $16^{2} \cdot 3$ .

Factor $256$ out of $768$ .

$x = \frac{- 16 \pm i \cdot \sqrt{256 (3)}}{2 \cdot 1}$

Rewrite $256$ as $16^{2}$ .

$x = \frac{- 16 \pm i \cdot \sqrt{16^{2} \cdot 3}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{- 16 \pm i \cdot (16 \sqrt{3})}{2 \cdot 1}$

Move $16$ to the left of $i$ .

$x = \frac{- 16 \pm 16 i \sqrt{3}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 16 \pm 16 i \sqrt{3}}{2}$

Simplify $\frac{- 16 \pm 16 i \sqrt{3}}{2}$ .

$x = - 8 \pm 8 i \sqrt{3}$

The final answer is the combination of both solutions.

$x = - 8 + 8 i \sqrt{3}, - 8 - 8 i \sqrt{3}$

The final solution is all the values that make $(x - 16) (x^{2} + 16 x + 256) = 0$ true.

$x = 16, - 8 + 8 i \sqrt{3}, - 8 - 8 i \sqrt{3}$

Solve for x 2 log base 4 of 64=3 log base 4 of x

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