Algebra Examples

$3^{165 x - 1} = {(\sqrt{k})}^{x}$

Step 1

Take the log of both sides of the equation.

$\ln (3^{165 x - 1}) = \ln ({\sqrt{k}}^{x})$

Step 2

Expand $\ln (3^{165 x - 1})$ by moving $165 x - 1$ outside the logarithm.

$(165 x - 1) \ln (3) = \ln ({\sqrt{k}}^{x})$

Step 3

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

$(165 x - 1) \ln (3) = \ln ({(k^{\frac{1}{2}})}^{x})$

Step 4

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

$(165 x - 1) \ln (3) = x \ln (k^{\frac{1}{2}})$

Step 5

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

$(165 x - 1) \ln (3) = x (\frac{1}{2} \ln (k))$

Step 6

Combine $\frac{1}{2}$ and $x$ .

$(165 x - 1) \ln (3) = \frac{x}{2} \ln (k)$

Step 7

Combine $\frac{x}{2}$ and $\ln (k)$ .

$(165 x - 1) \ln (3) = \frac{x \ln (k)}{2}$

Step 8

Solve the equation for $x$ .

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Step 8.1

Multiply both sides by $2$ .

$(165 x - 1) \ln (3) \cdot 2 = \frac{x \ln (k)}{2} \cdot 2$

Step 8.2

Simplify.

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Step 8.2.1

Simplify the left side.

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Step 8.2.1.1

Simplify $(165 x - 1) \ln (3) \cdot 2$ .

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Step 8.2.1.1.1

Apply the distributive property.

$(165 x \ln (3) - 1 \ln (3)) \cdot 2 = \frac{x \ln (k)}{2} \cdot 2$

Step 8.2.1.1.2

Rewrite $- 1 \ln (3)$ as $- \ln (3)$ .

$(165 x \ln (3) - \ln (3)) \cdot 2 = \frac{x \ln (k)}{2} \cdot 2$

Step 8.2.1.1.3

Apply the distributive property.

$165 x \ln (3) \cdot 2 - \ln (3) \cdot 2 = \frac{x \ln (k)}{2} \cdot 2$

Step 8.2.1.1.4

Multiply.

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Step 8.2.1.1.4.1

Multiply $2$ by $165$ .

$330 x \ln (3) - \ln (3) \cdot 2 = \frac{x \ln (k)}{2} \cdot 2$

Step 8.2.1.1.4.2

Multiply $2$ by $- 1$ .

$330 x \ln (3) - 2 \ln (3) = \frac{x \ln (k)}{2} \cdot 2$

Step 8.2.2

Simplify the right side.

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Step 8.2.2.1

Cancel the common factor of $2$ .

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Step 8.2.2.1.1

Cancel the common factor.

$330 x \ln (3) - 2 \ln (3) = \frac{x \ln (k)}{2} \cdot 2$

Step 8.2.2.1.2

Rewrite the expression.

$330 x \ln (3) - 2 \ln (3) = x \ln (k)$

Step 8.3

Solve for $x$ .

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Step 8.3.1

Subtract $x \ln (k)$ from both sides of the equation.

$330 x \ln (3) - 2 \ln (3) - x \ln (k) = 0$

Step 8.3.2

Add $2 \ln (3)$ to both sides of the equation.

$330 x \ln (3) - x \ln (k) = 2 \ln (3)$

Step 8.3.3

Factor $x$ out of $330 x \ln (3) - x \ln (k)$ .

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Step 8.3.3.1

Factor $x$ out of $330 x \ln (3)$ .

$x (330 \ln (3)) - x \ln (k) = 2 \ln (3)$

Step 8.3.3.2

Factor $x$ out of $- x \ln (k)$ .

$x (330 \ln (3)) + x (- 1 \ln (k)) = 2 \ln (3)$

Step 8.3.3.3

Factor $x$ out of $x (330 \ln (3)) + x (- 1 \ln (k))$ .

$x (330 \ln (3) - 1 \ln (k)) = 2 \ln (3)$

Step 8.3.4

Rewrite $- 1 \ln (k)$ as $- \ln (k)$ .

$x (330 \ln (3) - \ln (k)) = 2 \ln (3)$

Step 8.3.5

Divide each term in $x (330 \ln (3) - \ln (k)) = 2 \ln (3)$ by $330 \ln (3) - \ln (k)$ and simplify.

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Step 8.3.5.1

Divide each term in $x (330 \ln (3) - \ln (k)) = 2 \ln (3)$ by $330 \ln (3) - \ln (k)$ .

$\frac{x (330 \ln (3) - \ln (k))}{330 \ln (3) - \ln (k)} = \frac{2 \ln (3)}{330 \ln (3) - \ln (k)}$

Step 8.3.5.2

Simplify the left side.

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Step 8.3.5.2.1

Cancel the common factor of $330 \ln (3) - \ln (k)$ .

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Step 8.3.5.2.1.1

Cancel the common factor.

$\frac{x (330 \ln (3) - \ln (k))}{330 \ln (3) - \ln (k)} = \frac{2 \ln (3)}{330 \ln (3) - \ln (k)}$

Step 8.3.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{2 \ln (3)}{330 \ln (3) - \ln (k)}$