Precalculus Examples

$\log (2) + 3 \log (x) = 5 \log (y) - \log (5) - 2 \log (z)$

Step 1

Simplify the left side.

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Step 1.1

Simplify $\log (2) + 3 \log (x)$ .

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Step 1.1.1

Simplify $3 \log (x)$ by moving $3$ inside the logarithm.

$\log (2) + \log (x^{3}) = 5 \log (y) - \log (5) - 2 \log (z)$

Step 1.1.2

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\log (2 x^{3}) = 5 \log (y) - \log (5) - 2 \log (z)$

Step 2

Simplify the right side.

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Step 2.1

Simplify $5 \log (y) - \log (5) - 2 \log (z)$ .

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Simplify $5 \log (y)$ by moving $5$ inside the logarithm.

$\log (2 x^{3}) = \log (y^{5}) - \log (5) - 2 \log (z)$

Step 2.1.1.2

Simplify $- 2 \log (z)$ by moving $2$ inside the logarithm.

$\log (2 x^{3}) = \log (y^{5}) - \log (5) - \log (z^{2})$

Step 2.1.2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log (2 x^{3}) = \log (\frac{y^{5}}{5}) - \log (z^{2})$

Step 2.1.3

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log (2 x^{3}) = \log (\frac{\frac{y^{5}}{5}}{z^{2}})$

Step 2.1.4

Multiply the numerator by the reciprocal of the denominator.

$\log (2 x^{3}) = \log (\frac{y^{5}}{5} \cdot \frac{1}{z^{2}})$

Step 2.1.5

Combine.

$\log (2 x^{3}) = \log (\frac{y^{5} \cdot 1}{5 z^{2}})$

Step 2.1.6

Multiply $y^{5}$ by $1$ .

$\log (2 x^{3}) = \log (\frac{y^{5}}{5 z^{2}})$

Step 3

For the equation to be equal, the argument of the logarithms on both sides of the equation must be equal.

$2 x^{3} = \frac{y^{5}}{5 z^{2}}$

Step 4

Solve for $x$ .

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Step 4.1

Divide each term in $2 x^{3} = \frac{y^{5}}{5 z^{2}}$ by $2$ and simplify.

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Step 4.1.1

Divide each term in $2 x^{3} = \frac{y^{5}}{5 z^{2}}$ by $2$ .

$\frac{2 x^{3}}{2} = \frac{\frac{y^{5}}{5 z^{2}}}{2}$

Step 4.1.2

Simplify the left side.

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Step 4.1.2.1

Cancel the common factor of $2$ .

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Step 4.1.2.1.1

Cancel the common factor.

$\frac{2 x^{3}}{2} = \frac{\frac{y^{5}}{5 z^{2}}}{2}$

Step 4.1.2.1.2

Divide $x^{3}$ by $1$ .

$x^{3} = \frac{\frac{y^{5}}{5 z^{2}}}{2}$

Step 4.1.3

Simplify the right side.

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Step 4.1.3.1

Multiply the numerator by the reciprocal of the denominator.

$x^{3} = \frac{y^{5}}{5 z^{2}} \cdot \frac{1}{2}$

Step 4.1.3.2

Combine.

$x^{3} = \frac{y^{5} \cdot 1}{5 z^{2} \cdot 2}$

Step 4.1.3.3

Multiply.

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Step 4.1.3.3.1

Multiply $y^{5}$ by $1$ .

$x^{3} = \frac{y^{5}}{5 z^{2} \cdot 2}$

Step 4.1.3.3.2

Multiply $2$ by $5$ .

$x^{3} = \frac{y^{5}}{10 z^{2}}$

Step 4.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \sqrt[3]{\frac{y^{5}}{10 z^{2}}}$

Step 4.3

Simplify $\sqrt[3]{\frac{y^{5}}{10 z^{2}}}$ .

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Step 4.3.1

Rewrite $\sqrt[3]{\frac{y^{5}}{10 z^{2}}}$ as $\frac{\sqrt[3]{y^{5}}}{\sqrt[3]{10 z^{2}}}$ .

$x = \frac{\sqrt[3]{y^{5}}}{\sqrt[3]{10 z^{2}}}$

Step 4.3.2

Simplify the numerator.

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Step 4.3.2.1

Factor out $y^{3}$ .

$x = \frac{\sqrt[3]{y^{3} y^{2}}}{\sqrt[3]{10 z^{2}}}$

Step 4.3.2.2

Pull terms out from under the radical.

$x = \frac{y \sqrt[3]{y^{2}}}{\sqrt[3]{10 z^{2}}}$

Step 4.3.3

Multiply $\frac{y \sqrt[3]{y^{2}}}{\sqrt[3]{10 z^{2}}}$ by $\frac{{\sqrt[3]{10 z^{2}}}^{2}}{{\sqrt[3]{10 z^{2}}}^{2}}$ .

$x = \frac{y \sqrt[3]{y^{2}}}{\sqrt[3]{10 z^{2}}} \cdot \frac{{\sqrt[3]{10 z^{2}}}^{2}}{{\sqrt[3]{10 z^{2}}}^{2}}$

Step 4.3.4

Combine and simplify the denominator.

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Step 4.3.4.1

Multiply $\frac{y \sqrt[3]{y^{2}}}{\sqrt[3]{10 z^{2}}}$ by $\frac{{\sqrt[3]{10 z^{2}}}^{2}}{{\sqrt[3]{10 z^{2}}}^{2}}$ .

$x = \frac{y \sqrt[3]{y^{2}} {\sqrt[3]{10 z^{2}}}^{2}}{\sqrt[3]{10 z^{2}} {\sqrt[3]{10 z^{2}}}^{2}}$

Step 4.3.4.2

Raise $\sqrt[3]{10 z^{2}}$ to the power of $1$ .

$x = \frac{y \sqrt[3]{y^{2}} {\sqrt[3]{10 z^{2}}}^{2}}{{\sqrt[3]{10 z^{2}}}^{1} {\sqrt[3]{10 z^{2}}}^{2}}$

Step 4.3.4.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \frac{y \sqrt[3]{y^{2}} {\sqrt[3]{10 z^{2}}}^{2}}{{\sqrt[3]{10 z^{2}}}^{1 + 2}}$

Step 4.3.4.4

Add $1$ and $2$ .

$x = \frac{y \sqrt[3]{y^{2}} {\sqrt[3]{10 z^{2}}}^{2}}{{\sqrt[3]{10 z^{2}}}^{3}}$

Step 4.3.4.5

Rewrite ${\sqrt[3]{10 z^{2}}}^{3}$ as $10 z^{2}$ .

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Step 4.3.4.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{10 z^{2}}$ as ${(10 z^{2})}^{\frac{1}{3}}$ .

$x = \frac{y \sqrt[3]{y^{2}} {\sqrt[3]{10 z^{2}}}^{2}}{{({(10 z^{2})}^{\frac{1}{3}})}^{3}}$

Step 4.3.4.5.2

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

$x = \frac{y \sqrt[3]{y^{2}} {\sqrt[3]{10 z^{2}}}^{2}}{{(10 z^{2})}^{\frac{1}{3} \cdot 3}}$

Step 4.3.4.5.3

Combine $\frac{1}{3}$ and $3$ .

$x = \frac{y \sqrt[3]{y^{2}} {\sqrt[3]{10 z^{2}}}^{2}}{{(10 z^{2})}^{\frac{3}{3}}}$

Step 4.3.4.5.4

Cancel the common factor of $3$ .

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Step 4.3.4.5.4.1

Cancel the common factor.

$x = \frac{y \sqrt[3]{y^{2}} {\sqrt[3]{10 z^{2}}}^{2}}{{(10 z^{2})}^{\frac{3}{3}}}$

Step 4.3.4.5.4.2

Rewrite the expression.

$x = \frac{y \sqrt[3]{y^{2}} {\sqrt[3]{10 z^{2}}}^{2}}{{(10 z^{2})}^{1}}$

Step 4.3.4.5.5

Simplify.

$x = \frac{y \sqrt[3]{y^{2}} {\sqrt[3]{10 z^{2}}}^{2}}{10 z^{2}}$

Step 4.3.5

Simplify the numerator.

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Step 4.3.5.1

Rewrite ${\sqrt[3]{10 z^{2}}}^{2}$ as $\sqrt[3]{{(10 z^{2})}^{2}}$ .

$x = \frac{y \sqrt[3]{y^{2}} \sqrt[3]{{(10 z^{2})}^{2}}}{10 z^{2}}$

Step 4.3.5.2

Apply the product rule to $10 z^{2}$ .

$x = \frac{y \sqrt[3]{y^{2}} \sqrt[3]{10^{2} {(z^{2})}^{2}}}{10 z^{2}}$

Step 4.3.5.3

Raise $10$ to the power of $2$ .

$x = \frac{y \sqrt[3]{y^{2}} \sqrt[3]{100 {(z^{2})}^{2}}}{10 z^{2}}$

Step 4.3.5.4

Multiply the exponents in ${(z^{2})}^{2}$ .

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Step 4.3.5.4.1

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

$x = \frac{y \sqrt[3]{y^{2}} \sqrt[3]{100 z^{2 \cdot 2}}}{10 z^{2}}$

Step 4.3.5.4.2

Multiply $2$ by $2$ .

$x = \frac{y \sqrt[3]{y^{2}} \sqrt[3]{100 z^{4}}}{10 z^{2}}$

Step 4.3.5.5

Rewrite $100 z^{4}$ as $z^{3} \cdot (100 z)$ .

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Step 4.3.5.5.1

Factor out $z^{3}$ .

$x = \frac{y \sqrt[3]{y^{2}} \sqrt[3]{100 (z^{3} z)}}{10 z^{2}}$

Step 4.3.5.5.2

Reorder $100$ and $z^{3}$ .

$x = \frac{y \sqrt[3]{y^{2}} \sqrt[3]{z^{3} \cdot 100 z}}{10 z^{2}}$

Step 4.3.5.5.3

Add parentheses.

$x = \frac{y \sqrt[3]{y^{2}} \sqrt[3]{z^{3} \cdot (100 z)}}{10 z^{2}}$

Step 4.3.5.6

Pull terms out from under the radical.

$x = \frac{y \sqrt[3]{y^{2}} z \sqrt[3]{100 z}}{10 z^{2}}$

Step 4.3.5.7

Combine using the product rule for radicals.

$x = \frac{y z \sqrt[3]{y^{2} \cdot 100 z}}{10 z^{2}}$

Step 4.3.6

Reduce the expression by cancelling the common factors.

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Step 4.3.6.1

Cancel the common factor of $z$ and $z^{2}$ .

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Step 4.3.6.1.1

Factor $z$ out of $y z \sqrt[3]{y^{2} \cdot 100 z}$ .

$x = \frac{z (y \sqrt[3]{y^{2} \cdot 100 z})}{10 z^{2}}$

Step 4.3.6.1.2

Cancel the common factors.

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Step 4.3.6.1.2.1

Factor $z$ out of $10 z^{2}$ .

$x = \frac{z (y \sqrt[3]{y^{2} \cdot 100 z})}{z (10 z)}$

Step 4.3.6.1.2.2

Cancel the common factor.

$x = \frac{z (y \sqrt[3]{y^{2} \cdot 100 z})}{z (10 z)}$

Step 4.3.6.1.2.3

Rewrite the expression.

$x = \frac{y \sqrt[3]{y^{2} \cdot 100 z}}{10 z}$

Step 4.3.6.2

Move $100$ to the left of $y^{2}$ .

$x = \frac{y \sqrt[3]{100 y^{2} z}}{10 z}$