Calculus Examples

$e^{\sqrt{t}} = x^{6}$

Step 1

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

$\ln (e^{\sqrt{t}}) = \ln (x^{6})$

Step 2

Expand the left side.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{t}$ as $t^{\frac{1}{2}}$ .

$\ln (e^{t^{\frac{1}{2}}}) = \ln (x^{6})$

Step 2.2

Expand $\ln (e^{t^{\frac{1}{2}}})$ by moving $t^{\frac{1}{2}}$ outside the logarithm.

$t^{\frac{1}{2}} \ln (e) = \ln (x^{6})$

Step 2.3

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

$t^{\frac{1}{2}} \cdot 1 = \ln (x^{6})$

Step 2.4

Multiply $t^{\frac{1}{2}}$ by $1$ .

$t^{\frac{1}{2}} = \ln (x^{6})$

Step 3

Raise each side of the equation to the power of $2$ to eliminate the fractional exponent on the left side.

${(t^{\frac{1}{2}})}^{2} = \ln^{2} (x^{6})$

Step 4

Simplify the left side.

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Step 4.1

Simplify ${(t^{\frac{1}{2}})}^{2}$ .

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Step 4.1.1

Multiply the exponents in ${(t^{\frac{1}{2}})}^{2}$ .

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Step 4.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$t^{\frac{1}{2} \cdot 2} = \ln^{2} (x^{6})$

Step 4.1.1.2

Cancel the common factor of $2$ .

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Step 4.1.1.2.1

Cancel the common factor.

$t^{\frac{1}{2} \cdot 2} = \ln^{2} (x^{6})$

Step 4.1.1.2.2

Rewrite the expression.

$t^{1} = \ln^{2} (x^{6})$

Step 4.1.2

Simplify.

$t = \ln^{2} (x^{6})$