Precalculus Examples

$\frac{1}{3} \cdot \log (c^{x}) + 2 \log (c^{y}) - 5 \log (c^{x})$

Step 1

Simplify each term.

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Step 1.1

Simplify $\frac{1}{3} \log (c^{x})$ by moving $\frac{1}{3}$ inside the logarithm.

$\log ({(c^{x})}^{\frac{1}{3}}) + 2 \log (c^{y}) - 5 \log (c^{x})$

Step 1.2

Multiply the exponents in ${(c^{x})}^{\frac{1}{3}}$ .

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Step 1.2.1

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

$\log (c^{x \frac{1}{3}}) + 2 \log (c^{y}) - 5 \log (c^{x})$

Step 1.2.2

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

$\log (c^{\frac{x}{3}}) + 2 \log (c^{y}) - 5 \log (c^{x})$

Step 1.3

Simplify $2 \log (c^{y})$ by moving $2$ inside the logarithm.

$\log (c^{\frac{x}{3}}) + \log ({(c^{y})}^{2}) - 5 \log (c^{x})$

Step 1.4

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

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Step 1.4.1

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

$\log (c^{\frac{x}{3}}) + \log (c^{y \cdot 2}) - 5 \log (c^{x})$

Step 1.4.2

Move $2$ to the left of $y$ .

$\log (c^{\frac{x}{3}}) + \log (c^{2 y}) - 5 \log (c^{x})$

Step 1.5

Simplify $- 5 \log (c^{x})$ by moving $5$ inside the logarithm.

$\log (c^{\frac{x}{3}}) + \log (c^{2 y}) - \log ({(c^{x})}^{5})$

Step 1.6

Multiply the exponents in ${(c^{x})}^{5}$ .

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Step 1.6.1

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

$\log (c^{\frac{x}{3}}) + \log (c^{2 y}) - \log (c^{x \cdot 5})$

Step 1.6.2

Move $5$ to the left of $x$ .

$\log (c^{\frac{x}{3}}) + \log (c^{2 y}) - \log (c^{5 x})$

Step 2

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

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

Step 3

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

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

Step 4

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

$\log (\frac{c^{\frac{x}{3} + 2 y}}{c^{5 x}})$

Step 5

Cancel the common factor of $c^{\frac{x}{3} + 2 y}$ and $c^{5 x}$ .

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Step 5.1

Factor $c^{5 x}$ out of $c^{\frac{x}{3} + 2 y}$ .

$\log (\frac{c^{5 x} c^{2 y + \frac{x - 5 x \cdot 3}{3}}}{c^{5 x}})$

Step 5.2

Cancel the common factors.

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Step 5.2.1

Multiply by $1$ .

$\log (\frac{c^{5 x} c^{2 y + \frac{x - 5 x \cdot 3}{3}}}{c^{5 x} \cdot 1})$

Step 5.2.2

Cancel the common factor.

$\log (\frac{c^{5 x} c^{2 y + \frac{x - 5 x \cdot 3}{3}}}{c^{5 x} \cdot 1})$

Step 5.2.3

Rewrite the expression.

$\log (\frac{c^{2 y + \frac{x - 5 x \cdot 3}{3}}}{1})$

Step 5.2.4

Divide $c^{2 y + \frac{x - 5 x \cdot 3}{3}}$ by $1$ .

$\log (c^{2 y + \frac{x - 5 x \cdot 3}{3}})$

Step 6

Simplify each term.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Factor $x$ out of $x - 5 x \cdot 3$ .

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Step 6.1.1.1

Raise $x$ to the power of $1$ .

$\log (c^{2 y + \frac{x^{1} - 5 x \cdot 3}{3}})$

Step 6.1.1.2

Factor $x$ out of $x^{1}$ .

$\log (c^{2 y + \frac{x \cdot 1 - 5 x \cdot 3}{3}})$

Step 6.1.1.3

Factor $x$ out of $- 5 x \cdot 3$ .

$\log (c^{2 y + \frac{x \cdot 1 + x (- 5 \cdot 3)}{3}})$

Step 6.1.1.4

Factor $x$ out of $x \cdot 1 + x (- 5 \cdot 3)$ .

$\log (c^{2 y + \frac{x (1 - 5 \cdot 3)}{3}})$

Step 6.1.2

Multiply $- 5$ by $3$ .

$\log (c^{2 y + \frac{x (1 - 15)}{3}})$

Step 6.1.3

Subtract $15$ from $1$ .

$\log (c^{2 y + \frac{x \cdot - 14}{3}})$

Step 6.2

Move $- 14$ to the left of $x$ .

$\log (c^{2 y + \frac{- 14 \cdot x}{3}})$

Step 6.3

Move the negative in front of the fraction.

$\log (c^{2 y - \frac{14 x}{3}})$