Calculus Examples

$\ln (x^{7}) - 4 \ln (x) = 1$

Step 1

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

$\ln (x^{7}) - 4 \ln (x) = 1$

Step 2

Simplify the left side.

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Step 2.1

Simplify $\ln (x^{7}) - 4 \ln (x)$ .

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Step 2.1.1

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

$\ln (x^{7}) - \ln (x^{4}) = 1$

Step 2.1.2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\ln (\frac{x^{7}}{x^{4}}) = 1$

Step 2.1.3

Cancel the common factor of $x^{7}$ and $x^{4}$ .

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Step 2.1.3.1

Factor $x^{4}$ out of $x^{7}$ .

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

Step 2.1.3.2

Cancel the common factors.

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Step 2.1.3.2.1

Multiply by $1$ .

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

Step 2.1.3.2.2

Cancel the common factor.

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

Step 2.1.3.2.3

Rewrite the expression.

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

Step 2.1.3.2.4

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

$\ln (x^{3}) = 1$

Step 3

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

$e^{\ln (x^{3})} = e^{1}$

Step 4

Rewrite $\ln (x^{3}) = 1$ 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 $b^{y} = x$ .

$e^{1} = x^{3}$

Step 5

Solve for $x$ .

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Step 5.1

Rewrite the equation as $x^{3} = e^{1}$ .

$x^{3} = e$

Step 5.2

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

$x = \sqrt[3]{e^{1}}$

Step 5.3

Simplify.

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 1.39561242 \dots$