Calculus Examples

$\sqrt{3^{x}} = 50$

Step 1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{3^{x}}}^{2} = 50^{2}$

Step 2

Simplify each side of the equation.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3^{x}}$ as $3^{\frac{x}{2}}$ .

${(3^{\frac{x}{2}})}^{2} = 50^{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Multiply the exponents in ${(3^{\frac{x}{2}})}^{2}$ .

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Step 2.2.1.1

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

$3^{\frac{x}{2} \cdot 2} = 50^{2}$

Step 2.2.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.2.1

Cancel the common factor.

$3^{\frac{x}{2} \cdot 2} = 50^{2}$

Step 2.2.1.2.2

Rewrite the expression.

$3^{x} = 50^{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Raise $50$ to the power of $2$ .

$3^{x} = 2500$

Step 3

Solve for $x$ .

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Step 3.1

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

$\ln (3^{x}) = \ln (2500)$

Step 3.2

Expand $\ln (3^{x})$ by moving $x$ outside the logarithm.

$x \ln (3) = \ln (2500)$

Step 3.3

Divide each term in $x \ln (3) = \ln (2500)$ by $\ln (3)$ and simplify.

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Step 3.3.1

Divide each term in $x \ln (3) = \ln (2500)$ by $\ln (3)$ .

$\frac{x \ln (3)}{\ln (3)} = \frac{\ln (2500)}{\ln (3)}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $\ln (3)$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{x \ln (3)}{\ln (3)} = \frac{\ln (2500)}{\ln (3)}$

Step 3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{\ln (2500)}{\ln (3)}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\ln (2500)}{\ln (3)}$

Decimal Form:

$x = 7.12175359 \dots$