Precalculus Examples

$\log_{6} (x + 2) + \frac{1}{6} \cdot \log_{6} (x)$

Step 1

Simplify $\frac{1}{6} \log_{6} (x)$ by moving $\frac{1}{6}$ inside the logarithm.

$\log_{6} (x + 2) + \log_{6} (x^{\frac{1}{6}})$

Step 2

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

$\log_{6} ((x + 2) x^{\frac{1}{6}})$

Step 3

Apply the distributive property.

$\log_{6} (x \cdot x^{\frac{1}{6}} + 2 x^{\frac{1}{6}})$

Step 4

Multiply $x$ by $x^{\frac{1}{6}}$ by adding the exponents.

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Step 4.1

Multiply $x$ by $x^{\frac{1}{6}}$ .

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Step 4.1.1

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

$\log_{6} (x^{1} x^{\frac{1}{6}} + 2 x^{\frac{1}{6}})$

Step 4.1.2

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

$\log_{6} (x^{1 + \frac{1}{6}} + 2 x^{\frac{1}{6}})$

Step 4.2

Write $1$ as a fraction with a common denominator.

$\log_{6} (x^{\frac{6}{6} + \frac{1}{6}} + 2 x^{\frac{1}{6}})$

Step 4.3

Combine the numerators over the common denominator.

$\log_{6} (x^{\frac{6 + 1}{6}} + 2 x^{\frac{1}{6}})$

Step 4.4

Add $6$ and $1$ .

$\log_{6} (x^{\frac{7}{6}} + 2 x^{\frac{1}{6}})$