Algebra Examples

$e^{\frac{1}{4} x} = | 4 x |$

Step 1

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

$\ln (e^{\frac{1}{4} x}) = \ln (| 4 x |)$

Step 2

Expand the left side.

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Step 2.1

Expand $\ln (e^{\frac{1}{4} x})$ by moving $\frac{1}{4} x$ outside the logarithm.

$\frac{1}{4} x \ln (e) = \ln (| 4 x |)$

Step 2.2

The natural logarithm of $e$ is $1$ .

$\frac{1}{4} x \cdot 1 = \ln (| 4 x |)$

Step 2.3

Combine $\frac{1}{4}$ and $x$ .

$\frac{x}{4} \cdot 1 = \ln (| 4 x |)$

Step 2.4

Multiply $\frac{x}{4}$ by $1$ .

$\frac{x}{4} = \ln (| 4 x |)$

Step 3

Simplify the right side.

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Step 3.1

Remove non-negative terms from the absolute value.

$\frac{x}{4} = \ln (4 | x |)$

Step 4

Multiply both sides by $4$ .

$\frac{x}{4} \cdot 4 = \ln (4 | x |) \cdot 4$

Step 5

Simplify.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Cancel the common factor of $4$ .

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Step 5.1.1.1

Cancel the common factor.

$\frac{x}{4} \cdot 4 = \ln (4 | x |) \cdot 4$

Step 5.1.1.2

Rewrite the expression.

$x = \ln (4 | x |) \cdot 4$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $\ln (4 | x |) \cdot 4$ .

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Step 5.2.1.1

Multiply $\ln (4 | x |) \cdot 4$ .

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Step 5.2.1.1.1

Reorder $\ln (4 | x |)$ and $4$ .

$x = 4 \cdot \ln (4 | x |)$

Step 5.2.1.1.2

Simplify $4 \cdot \ln (4 | x |)$ by moving $4$ inside the logarithm.

$x = \ln ({(4 | x |)}^{4})$

Step 5.2.1.2

Apply the product rule to $4 | x |$ .

$x = \ln (4^{4} {| x |}^{4})$

Step 5.2.1.3

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

$x = \ln (256 {| x |}^{4})$

Step 5.2.1.4

Remove the absolute value in ${| x |}^{4}$ because exponentiations with even powers are always positive.

$x = \ln (256 x^{4})$

Step 6

Expand the left side.

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Step 6.1

Expand $\ln (e^{\frac{1}{4} x})$ by moving $\frac{1}{4} x$ outside the logarithm.

$\frac{1}{4} x \ln (e) = \ln (| 4 x |)$

Step 6.2

The natural logarithm of $e$ is $1$ .

$\frac{1}{4} x \cdot 1 = \ln (| 4 x |)$

Step 6.3

Combine $\frac{1}{4}$ and $x$ .

$\frac{x}{4} \cdot 1 = \ln (| 4 x |)$

Step 6.4

Multiply $\frac{x}{4}$ by $1$ .

$\frac{x}{4} = \ln (| 4 x |)$