Calculus Examples

$\ln (\ln (3 x)) = 0$

Step 1

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

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

Step 2

Rewrite $\ln (\ln (3 x)) = 0$ 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 $b^{y} = x$ .

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

Step 3

Solve for $x$ .

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Step 3.1

Rewrite the equation as $\ln (3 x) = e^{0}$ .

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

Step 3.2

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

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

Step 3.3

Rewrite $\ln (3 x) = e^{0}$ 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 $b^{y} = x$ .

$e^{e^{0}} = 3 x$

Step 3.4

Solve for $x$ .

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Step 3.4.1

Rewrite the equation as $3 x = e^{e^{0}}$ .

$3 x = e^{e^{0}}$

Step 3.4.2

Anything raised to $0$ is $1$ .

$3 x = e^{1}$

Step 3.4.3

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

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Step 3.4.3.1

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

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

Step 3.4.3.2

Simplify the left side.

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Step 3.4.3.2.1

Cancel the common factor of $3$ .

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Step 3.4.3.2.1.1

Cancel the common factor.

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

Step 3.4.3.2.1.2

Divide $x$ by $1$ .

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

Step 3.4.3.3

Simplify the right side.

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Step 3.4.3.3.1

Simplify.

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

Step 4

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.90609394 \dots$