Calculus Examples

$2 x^{- 3} \ln (x) - 3 x^{- 3} = 0$

Step 1

Simplify each term.

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Step 1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$2 \frac{1}{x^{3}} \ln (x) - 3 x^{- 3} = 0$

Step 1.2

Combine $2$ and $\frac{1}{x^{3}}$ .

$\frac{2}{x^{3}} \ln (x) - 3 x^{- 3} = 0$

Step 1.3

Combine $\frac{2}{x^{3}}$ and $\ln (x)$ .

$\frac{2 \ln (x)}{x^{3}} - 3 x^{- 3} = 0$

Step 1.4

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{2 \ln (x)}{x^{3}} - 3 \frac{1}{x^{3}} = 0$

Step 1.5

Combine $- 3$ and $\frac{1}{x^{3}}$ .

$\frac{2 \ln (x)}{x^{3}} + \frac{- 3}{x^{3}} = 0$

Step 1.6

Move the negative in front of the fraction.

$\frac{2 \ln (x)}{x^{3}} - \frac{3}{x^{3}} = 0$

Step 2

Add $\frac{3}{x^{3}}$ to both sides of the equation.

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

Step 3

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$2 \ln (x) = 3$

Step 4

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

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Step 4.1

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

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

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $2$ .

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Step 4.2.1.1

Cancel the common factor.

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

Step 4.2.1.2

Divide $\ln (x)$ by $1$ .

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

Step 5

To solve for $x$ , rewrite the equation using properties of logarithms.

$e^{\ln (x)} = e^{\frac{3}{2}}$

Step 6

Rewrite $\ln (x) = \frac{3}{2}$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{\frac{3}{2}} = x$

Step 7

Rewrite the equation as $x = e^{\frac{3}{2}}$ .

$x = e^{\frac{3}{2}}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = e^{\frac{3}{2}}$

Decimal Form:

$x = 4.48168907 \dots$