Precalculus Examples

$\frac{1}{2} \cdot \log (8 x + 1) - (\log (x + 1) - \frac{3}{2} \cdot \log (8 x + 1))$

Step 1

Simplify each term.

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Step 1.1

Simplify $\frac{1}{2} \log (8 x + 1)$ by moving $\frac{1}{2}$ inside the logarithm.

$\log ({(8 x + 1)}^{\frac{1}{2}}) - (\log (x + 1) - \frac{3}{2} \cdot \log (8 x + 1))$

Step 1.2

Multiply $- \frac{3}{2} \log (8 x + 1)$ .

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Step 1.2.1

Reorder $\log (8 x + 1)$ and $\frac{3}{2}$ .

$\log ({(8 x + 1)}^{\frac{1}{2}}) - (\log (x + 1) - (\frac{3}{2} \log (8 x + 1)))$

Step 1.2.2

Simplify $\frac{3}{2} \log (8 x + 1)$ by moving $\frac{3}{2}$ inside the logarithm.

$\log ({(8 x + 1)}^{\frac{1}{2}}) - (\log (x + 1) - \log ({(8 x + 1)}^{\frac{3}{2}}))$

Step 1.3

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

$\log ({(8 x + 1)}^{\frac{1}{2}}) - \log (\frac{x + 1}{{(8 x + 1)}^{\frac{3}{2}}})$

Step 2

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

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

Step 3

Multiply the numerator by the reciprocal of the denominator.

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

Step 4

Multiply ${(8 x + 1)}^{\frac{1}{2}} \frac{{(8 x + 1)}^{\frac{3}{2}}}{x + 1}$ .

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Step 4.1

Combine ${(8 x + 1)}^{\frac{1}{2}}$ and $\frac{{(8 x + 1)}^{\frac{3}{2}}}{x + 1}$ .

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

Step 4.2

Multiply ${(8 x + 1)}^{\frac{1}{2}}$ by ${(8 x + 1)}^{\frac{3}{2}}$ by adding the exponents.

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Step 4.2.1

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

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

Step 4.2.2

Combine the numerators over the common denominator.

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

Step 4.2.3

Add $1$ and $3$ .

$\log (\frac{{(8 x + 1)}^{\frac{4}{2}}}{x + 1})$

Step 4.2.4

Divide $4$ by $2$ .

$\log (\frac{{(8 x + 1)}^{2}}{x + 1})$