Precalculus Examples

$\log_{\sqrt{3} \cdot 3} (8)$

Step 1

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

$\log_{3^{\frac{1}{2}} \cdot 3} (8)$

Step 2

Multiply $3^{\frac{1}{2}}$ by $3$ by adding the exponents.

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Step 2.1

Multiply $3^{\frac{1}{2}}$ by $3$ .

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Step 2.1.1

Raise $3$ to the power of $1$ .

$\log_{3^{\frac{1}{2}} \cdot 3^{1}} (8)$

Step 2.1.2

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

$\log_{3^{\frac{1}{2} + 1}} (8)$

Step 2.2

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

$\log_{3^{\frac{1}{2} + \frac{2}{2}}} (8)$

Step 2.3

Combine the numerators over the common denominator.

$\log_{3^{\frac{1 + 2}{2}}} (8)$

Step 2.4

Add $1$ and $2$ .

$\log_{3^{\frac{3}{2}}} (8)$

Step 3

Rewrite $\log_{3^{\frac{3}{2}}} (8)$ as $\log_{3^{\frac{3}{2}}} (2^{3})$ .

$\log_{3^{\frac{3}{2}}} (2^{3})$

Step 4

Expand $\log_{3^{\frac{3}{2}}} (2^{3})$ by moving $3$ outside the logarithm.

$3 \log_{3^{\frac{3}{2}}} (2)$