Precalculus Examples

$\log_{4} (\sqrt{5.5}) - \log_{4} (\sqrt{5.5} - 1)$

Step 1

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

$\log_{4} (\frac{\sqrt{5.5}}{\sqrt{5.5} - 1})$

Step 2

Multiply $\frac{\sqrt{5.5}}{\sqrt{5.5} - 1}$ by $\frac{\sqrt{5.5} + 1}{\sqrt{5.5} + 1}$ .

$\log_{4} (\frac{\sqrt{5.5}}{\sqrt{5.5} - 1} \cdot \frac{\sqrt{5.5} + 1}{\sqrt{5.5} + 1})$

Step 3

Multiply $\frac{\sqrt{5.5}}{\sqrt{5.5} - 1}$ by $\frac{\sqrt{5.5} + 1}{\sqrt{5.5} + 1}$ .

$\log_{4} (\frac{\sqrt{5.5} (\sqrt{5.5} + 1)}{(\sqrt{5.5} - 1) (\sqrt{5.5} + 1)})$

Step 4

Expand the denominator using the FOIL method.

$\log_{4} (\frac{\sqrt{5.5} (\sqrt{5.5} + 1)}{{\sqrt{5.5}}^{2} + \sqrt{5.5} - \sqrt{5.5} - 1})$

Step 5

Simplify.

$\log_{4} (\frac{\sqrt{5.5} (\sqrt{5.5} + 1)}{4.5})$

Step 6

Apply the distributive property.

$\log_{4} (\frac{\sqrt{5.5} \sqrt{5.5} + \sqrt{5.5} \cdot 1}{4.5})$

Step 7

Multiply $\sqrt{5.5} \sqrt{5.5}$ .

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Step 7.1

Raise $\sqrt{5.5}$ to the power of $1$ .

$\log_{4} (\frac{{\sqrt{5.5}}^{1} \sqrt{5.5} + \sqrt{5.5} \cdot 1}{4.5})$

Step 7.2

Raise $\sqrt{5.5}$ to the power of $1$ .

$\log_{4} (\frac{{\sqrt{5.5}}^{1} {\sqrt{5.5}}^{1} + \sqrt{5.5} \cdot 1}{4.5})$

Step 7.3

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

$\log_{4} (\frac{{\sqrt{5.5}}^{1 + 1} + \sqrt{5.5} \cdot 1}{4.5})$

Step 7.4

Add $1$ and $1$ .

$\log_{4} (\frac{{\sqrt{5.5}}^{2} + \sqrt{5.5} \cdot 1}{4.5})$

Step 8

Multiply $\sqrt{5.5}$ by $1$ .

$\log_{4} (\frac{{\sqrt{5.5}}^{2} + \sqrt{5.5}}{4.5})$

Step 9

Rewrite ${\sqrt{5.5}}^{2}$ as $5.5$ .

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Step 9.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5.5}$ as ${5.5}^{\frac{1}{2}}$ .

$\log_{4} (\frac{{({5.5}^{\frac{1}{2}})}^{2} + \sqrt{5.5}}{4.5})$

Step 9.2

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

$\log_{4} (\frac{{5.5}^{\frac{1}{2} \cdot 2} + \sqrt{5.5}}{4.5})$

Step 9.3

Combine $\frac{1}{2}$ and $2$ .

$\log_{4} (\frac{{5.5}^{\frac{2}{2}} + \sqrt{5.5}}{4.5})$

Step 9.4

Cancel the common factor of $2$ .

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Step 9.4.1

Cancel the common factor.

$\log_{4} (\frac{{5.5}^{\frac{2}{2}} + \sqrt{5.5}}{4.5})$

Step 9.4.2

Rewrite the expression.

$\log_{4} (\frac{{5.5}^{1} + \sqrt{5.5}}{4.5})$

Step 9.5

Evaluate the exponent.

$\log_{4} (\frac{5.5 + \sqrt{5.5}}{4.5})$

Step 10

Add $5.5$ and $\sqrt{5.5}$ .

$\log_{4} (\frac{7.84520787}{4.5})$

Step 11

Divide $7.84520787$ by $4.5$ .

$\log_{4} (1.74337952)$

Step 12

Log base $4$ of $1.74337952$ is approximately $0.40094333$ .

$0.40094333$