Algebra Examples

$3 = 20 {(\frac{1}{2})}^{2 x}$

Step 1

Rewrite the equation as $20 {(\frac{1}{2})}^{2 x} = 3$ .

$20 {(\frac{1}{2})}^{2 x} = 3$

Step 2

Divide each term in $20 {(\frac{1}{2})}^{2 x} = 3$ by $20$ and simplify.

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Step 2.1

Divide each term in $20 {(\frac{1}{2})}^{2 x} = 3$ by $20$ .

$\frac{20 {(\frac{1}{2})}^{2 x}}{20} = \frac{3}{20}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $20$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{20 {(\frac{1}{2})}^{2 x}}{20} = \frac{3}{20}$

Step 2.2.1.2

Divide ${(\frac{1}{2})}^{2 x}$ by $1$ .

${(\frac{1}{2})}^{2 x} = \frac{3}{20}$

Step 2.2.2

Simplify the expression.

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Step 2.2.2.1

Apply the product rule to $\frac{1}{2}$ .

$\frac{1^{2 x}}{2^{2 x}} = \frac{3}{20}$

Step 2.2.2.2

One to any power is one.

$\frac{1}{2^{2 x}} = \frac{3}{20}$

Step 3

Multiply both sides by $2^{2 x}$ .

$\frac{1}{2^{2 x}} \cdot 2^{2 x} = \frac{3}{20} \cdot 2^{2 x}$

Step 4

Simplify.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Cancel the common factor of $2^{2 x}$ .

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Step 4.1.1.1

Cancel the common factor.

$\frac{1}{2^{2 x}} \cdot 2^{2 x} = \frac{3}{20} \cdot 2^{2 x}$

Step 4.1.1.2

Rewrite the expression.

$1 = \frac{3}{20} \cdot 2^{2 x}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Combine $\frac{3}{20}$ and $2^{2 x}$ .

$1 = \frac{3 \cdot 2^{2 x}}{20}$

Step 5

Solve for $x$ .

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Step 5.1

Rewrite the equation as $\frac{3 \cdot 2^{2 x}}{20} = 1$ .

$\frac{3 \cdot 2^{2 x}}{20} = 1$

Step 5.2

Multiply both sides by $20$ .

$\frac{3 \cdot 2^{2 x}}{20} \cdot 20 = 1 \cdot 20$

Step 5.3

Simplify.

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Step 5.3.1

Simplify the left side.

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Step 5.3.1.1

Cancel the common factor of $20$ .

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Step 5.3.1.1.1

Cancel the common factor.

$\frac{3 \cdot 2^{2 x}}{20} \cdot 20 = 1 \cdot 20$

Step 5.3.1.1.2

Rewrite the expression.

$3 \cdot 2^{2 x} = 1 \cdot 20$

Step 5.3.2

Simplify the right side.

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Step 5.3.2.1

Multiply $20$ by $1$ .

$3 \cdot 2^{2 x} = 20$

Step 5.4

Solve for $x$ .

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Step 5.4.1

Divide each term in $3 \cdot 2^{2 x} = 20$ by $3$ and simplify.

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Step 5.4.1.1

Divide each term in $3 \cdot 2^{2 x} = 20$ by $3$ .

$\frac{3 \cdot 2^{2 x}}{3} = \frac{20}{3}$

Step 5.4.1.2

Simplify the left side.

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Step 5.4.1.2.1

Cancel the common factor of $3$ .

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Step 5.4.1.2.1.1

Cancel the common factor.

$\frac{3 \cdot 2^{2 x}}{3} = \frac{20}{3}$

Step 5.4.1.2.1.2

Divide $2^{2 x}$ by $1$ .

$2^{2 x} = \frac{20}{3}$

Step 5.4.2

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

$\ln (2^{2 x}) = \ln (\frac{20}{3})$

Step 5.4.3

Expand $\ln (2^{2 x})$ by moving $2 x$ outside the logarithm.

$2 x \ln (2) = \ln (\frac{20}{3})$

Step 5.4.4

Divide each term in $2 x \ln (2) = \ln (\frac{20}{3})$ by $2 \ln (2)$ and simplify.

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Step 5.4.4.1

Divide each term in $2 x \ln (2) = \ln (\frac{20}{3})$ by $2 \ln (2)$ .

$\frac{2 x \ln (2)}{2 \ln (2)} = \frac{\ln (\frac{20}{3})}{2 \ln (2)}$

Step 5.4.4.2

Simplify the left side.

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Step 5.4.4.2.1

Cancel the common factor of $2$ .

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Step 5.4.4.2.1.1

Cancel the common factor.

$\frac{2 x \ln (2)}{2 \ln (2)} = \frac{\ln (\frac{20}{3})}{2 \ln (2)}$

Step 5.4.4.2.1.2

Rewrite the expression.

$\frac{x \ln (2)}{\ln (2)} = \frac{\ln (\frac{20}{3})}{2 \ln (2)}$

Step 5.4.4.2.2

Cancel the common factor of $\ln (2)$ .

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Step 5.4.4.2.2.1

Cancel the common factor.

$\frac{x \ln (2)}{\ln (2)} = \frac{\ln (\frac{20}{3})}{2 \ln (2)}$

Step 5.4.4.2.2.2

Divide $x$ by $1$ .

$x = \frac{\ln (\frac{20}{3})}{2 \ln (2)}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\ln (\frac{20}{3})}{2 \ln (2)}$

Decimal Form:

$x = 1.36848279 \dots$