Precalculus Examples

$2 \log_{5} (x + 1) - \log_{5} (x + 1) + \log_{5} (x^{2})$

Step 1

Simplify $2 \log_{5} (x + 1)$ by moving $2$ inside the logarithm.

$\log_{5} ({(x + 1)}^{2}) - \log_{5} (x + 1) + \log_{5} (x^{2})$

Step 2

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

$\log_{5} (\frac{{(x + 1)}^{2}}{x + 1}) + \log_{5} (x^{2})$

Step 3

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

$\log_{5} (\frac{{(x + 1)}^{2}}{x + 1} x^{2})$

Step 4

Cancel the common factor of ${(x + 1)}^{2}$ and $x + 1$ .

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Step 4.1

Factor $x + 1$ out of ${(x + 1)}^{2}$ .

$\log_{5} (\frac{(x + 1) (x + 1)}{x + 1} x^{2})$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Multiply by $1$ .

$\log_{5} (\frac{(x + 1) (x + 1)}{(x + 1) \cdot 1} x^{2})$

Step 4.2.2

Cancel the common factor.

$\log_{5} (\frac{(x + 1) (x + 1)}{(x + 1) \cdot 1} x^{2})$

Step 4.2.3

Rewrite the expression.

$\log_{5} (\frac{x + 1}{1} x^{2})$

Step 4.2.4

Divide $x + 1$ by $1$ .

$\log_{5} ((x + 1) x^{2})$

Step 5

Apply the distributive property.

$\log_{5} (x \cdot x^{2} + 1 x^{2})$

Step 6

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 6.1

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

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Step 6.1.1

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

$\log_{5} (x^{1} x^{2} + 1 x^{2})$

Step 6.1.2

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

$\log_{5} (x^{1 + 2} + 1 x^{2})$

Step 6.2

Add $1$ and $2$ .

$\log_{5} (x^{3} + 1 x^{2})$

Step 7

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

$\log_{5} (x^{3} + x^{2})$