Trigonometry Examples

$y = \frac{1}{2} \cdot \log (x + 1) - \log (2 x + 10)$

Step 1

Rewrite the equation as $\frac{1}{2} \cdot \log (x + 1) - \log (2 x + 10) = y$ .

$\frac{1}{2} \cdot \log (x + 1) - \log (2 x + 10) = y$

Step 2

Simplify the left side.

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Step 2.1

Combine $\frac{1}{2}$ and $\log (x + 1)$ .

$\frac{\log (x + 1)}{2} - \log (2 x + 10) = y$

Step 3

To write $- \log (2 x + 10)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{\log (x + 1)}{2} - \log (2 x + 10) \cdot \frac{2}{2} = y$

Step 4

Combine $- \log (2 x + 10)$ and $\frac{2}{2}$ .

$\frac{\log (x + 1)}{2} + \frac{- \log (2 x + 10) \cdot 2}{2} = y$

Step 5

Combine the numerators over the common denominator.

$\frac{\log (x + 1) - \log (2 x + 10) \cdot 2}{2} = y$

Step 6

Multiply $2$ by $- 1$ .

$\frac{\log (x + 1) - 2 \log (2 x + 10)}{2} = y$

Step 7

Simplify the left side.

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Step 7.1

Simplify $\frac{\log (x + 1) - 2 \log (2 x + 10)}{2}$ .

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Step 7.1.1

Simplify the numerator.

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Step 7.1.1.1

Simplify $- 2 \log (2 x + 10)$ by moving $2$ inside the logarithm.

$\frac{\log (x + 1) - \log ({(2 x + 10)}^{2})}{2} = y$

Step 7.1.1.2

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

$\frac{\log (\frac{x + 1}{{(2 x + 10)}^{2}})}{2} = y$

Step 7.1.1.3

Simplify the denominator.

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Step 7.1.1.3.1

Factor $2$ out of $2 x + 10$ .

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Step 7.1.1.3.1.1

Factor $2$ out of $2 x$ .

$\frac{\log (\frac{x + 1}{{(2 (x) + 10)}^{2}})}{2} = y$

Step 7.1.1.3.1.2

Factor $2$ out of $10$ .

$\frac{\log (\frac{x + 1}{{(2 x + 2 \cdot 5)}^{2}})}{2} = y$

Step 7.1.1.3.1.3

Factor $2$ out of $2 x + 2 \cdot 5$ .

$\frac{\log (\frac{x + 1}{{(2 (x + 5))}^{2}})}{2} = y$

Step 7.1.1.3.2

Apply the product rule to $2 (x + 5)$ .

$\frac{\log (\frac{x + 1}{2^{2} {(x + 5)}^{2}})}{2} = y$

Step 7.1.1.3.3

Raise $2$ to the power of $2$ .

$\frac{\log (\frac{x + 1}{4 {(x + 5)}^{2}})}{2} = y$

Step 7.1.2

Rewrite $\frac{\log (\frac{x + 1}{4 {(x + 5)}^{2}})}{2}$ as $\frac{1}{2} \log (\frac{x + 1}{4 {(x + 5)}^{2}})$ .

$\frac{1}{2} \log (\frac{x + 1}{4 {(x + 5)}^{2}}) = y$

Step 7.1.3

Simplify $\frac{1}{2} \log (\frac{x + 1}{4 {(x + 5)}^{2}})$ by moving $\frac{1}{2}$ inside the logarithm.

$\log ({(\frac{x + 1}{4 {(x + 5)}^{2}})}^{\frac{1}{2}}) = y$

Step 7.1.4

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 7.1.4.1

Apply the product rule to $\frac{x + 1}{4 {(x + 5)}^{2}}$ .

$\log (\frac{{(x + 1)}^{\frac{1}{2}}}{{(4 {(x + 5)}^{2})}^{\frac{1}{2}}}) = y$

Step 7.1.4.2

Apply the product rule to $4 {(x + 5)}^{2}$ .

$\log (\frac{{(x + 1)}^{\frac{1}{2}}}{4^{\frac{1}{2}} {({(x + 5)}^{2})}^{\frac{1}{2}}}) = y$

Step 7.1.5

Simplify the denominator.

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Step 7.1.5.1

Rewrite $4$ as $2^{2}$ .

$\log (\frac{{(x + 1)}^{\frac{1}{2}}}{{(2^{2})}^{\frac{1}{2}} {({(x + 5)}^{2})}^{\frac{1}{2}}}) = y$

Step 7.1.5.2

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

$\log (\frac{{(x + 1)}^{\frac{1}{2}}}{2^{2 (\frac{1}{2})} {({(x + 5)}^{2})}^{\frac{1}{2}}}) = y$

Step 7.1.5.3

Cancel the common factor of $2$ .

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Step 7.1.5.3.1

Cancel the common factor.

$\log (\frac{{(x + 1)}^{\frac{1}{2}}}{2^{2 (\frac{1}{2})} {({(x + 5)}^{2})}^{\frac{1}{2}}}) = y$

Step 7.1.5.3.2

Rewrite the expression.

$\log (\frac{{(x + 1)}^{\frac{1}{2}}}{2^{1} {({(x + 5)}^{2})}^{\frac{1}{2}}}) = y$

Step 7.1.5.4

Evaluate the exponent.

$\log (\frac{{(x + 1)}^{\frac{1}{2}}}{2 {({(x + 5)}^{2})}^{\frac{1}{2}}}) = y$

Step 7.1.5.5

Multiply the exponents in ${({(x + 5)}^{2})}^{\frac{1}{2}}$ .

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Step 7.1.5.5.1

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

$\log (\frac{{(x + 1)}^{\frac{1}{2}}}{2 {(x + 5)}^{2 (\frac{1}{2})}}) = y$

Step 7.1.5.5.2

Cancel the common factor of $2$ .

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Step 7.1.5.5.2.1

Cancel the common factor.

$\log (\frac{{(x + 1)}^{\frac{1}{2}}}{2 {(x + 5)}^{2 (\frac{1}{2})}}) = y$

Step 7.1.5.5.2.2

Rewrite the expression.

$\log (\frac{{(x + 1)}^{\frac{1}{2}}}{2 {(x + 5)}^{1}}) = y$

Step 7.1.5.6

Simplify.

$\log (\frac{{(x + 1)}^{\frac{1}{2}}}{2 (x + 5)}) = y$