Trigonometry Examples

$\log (4 x) = \log (x^{4})$

Step 1

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

$4 x = x^{4}$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $x^{4}$ from both sides of the equation.

$4 x - x^{4} = 0$

Step 2.2

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

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Step 2.2.1

Factor $x$ out of $4 x$ .

$x \cdot 4 - x^{4} = 0$

Step 2.2.2

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

$x \cdot 4 + x (- x^{3}) = 0$

Step 2.2.3

Factor $x$ out of $x \cdot 4 + x (- x^{3})$ .

$x (4 - x^{3}) = 0$

Step 2.3

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

$x = 0$

$4 - x^{3} = 0$

Step 2.4

Set $x$ equal to $0$ .

$x = 0$

Step 2.5

Set $4 - x^{3}$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $4 - x^{3}$ equal to $0$ .

$4 - x^{3} = 0$

Step 2.5.2

Solve $4 - x^{3} = 0$ for $x$ .

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Step 2.5.2.1

Subtract $4$ from both sides of the equation.

$- x^{3} = - 4$

Step 2.5.2.2

Divide each term in $- x^{3} = - 4$ by $- 1$ and simplify.

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Step 2.5.2.2.1

Divide each term in $- x^{3} = - 4$ by $- 1$ .

$\frac{- x^{3}}{- 1} = \frac{- 4}{- 1}$

Step 2.5.2.2.2

Simplify the left side.

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Step 2.5.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x^{3}}{1} = \frac{- 4}{- 1}$

Step 2.5.2.2.2.2

Divide $x^{3}$ by $1$ .

$x^{3} = \frac{- 4}{- 1}$

Step 2.5.2.2.3

Simplify the right side.

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Step 2.5.2.2.3.1

Divide $- 4$ by $- 1$ .

$x^{3} = 4$

Step 2.5.2.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \sqrt[3]{4}$

Step 2.6

The final solution is all the values that make $x (4 - x^{3}) = 0$ true.

$x = 0, \sqrt[3]{4}$

Step 3

Exclude the solutions that do not make $\log (4 x) = \log (x^{4})$ true.

$x = \sqrt[3]{4}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \sqrt[3]{4}$

Decimal Form:

$x = 1.58740105 \dots$