Calculus Examples

$2 \log_{b} (x + 4) \log_{b} (2)$

Step 1

Simplify $2 \log_{b} (x + 4)$ by moving $2$ inside the logarithm.

$\log_{b} ({(x + 4)}^{2}) \log_{b} (2)$

Step 2

Rewrite $\log_{b} ({(x + 4)}^{2})$ using the change of base formula.

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Step 2.1

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} \log_{b} (2)$

Step 2.2

Substitute in values for the variables in the change of base formula, using $b = 10$ .

$\frac{\log ({(x + 4)}^{2})}{\log (b)} \log_{b} (2)$

Step 3

Rewrite $\log_{b} (2)$ using the change of base formula.

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Step 3.1

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$\frac{\log ({(x + 4)}^{2})}{\log (b)} \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}$

Step 3.2

Substitute in values for the variables in the change of base formula, using $b = 10$ .

$\frac{\log ({(x + 4)}^{2})}{\log (b)} \cdot \frac{\log (2)}{\log (b)}$

Step 4

Combine.

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

Step 5

Simplify the denominator.

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Step 5.1

Raise $\log (b)$ to the power of $1$ .

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

Step 5.2

Raise $\log (b)$ to the power of $1$ .

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

Step 5.3

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

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

Step 5.4

Add $1$ and $1$ .

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