Precalculus Examples

$\frac{5 \log_{7} (x + 1)}{3 \log_{7} (x - 2) \log_{7} (x - 1)}$

Step 1

Simplify $5 \log_{7} (x + 1)$ by moving $5$ inside the logarithm.

$\frac{\log_{7} ({(x + 1)}^{5})}{3 \log_{7} (x - 2) \log_{7} (x - 1)}$

Step 2

Simplify $3 \log_{7} (x - 2)$ by moving $3$ inside the logarithm.

$\frac{\log_{7} ({(x + 1)}^{5})}{\log_{7} ({(x - 2)}^{3}) \log_{7} (x - 1)}$

Step 3

Rewrite $\log_{7} ({(x + 1)}^{5})$ 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 $0$ .

$\frac{\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}}{\log_{7} ({(x - 2)}^{3}) \log_{7} (x - 1)}$

Step 3.2

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

$\frac{\frac{\log ({(x + 1)}^{5})}{\log (7)}}{\log_{7} ({(x - 2)}^{3}) \log_{7} (x - 1)}$

Step 4

Simplify the denominator.

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Step 4.1

Rewrite $\log_{7} ({(x - 2)}^{3})$ using the change of base formula.

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Step 4.1.1

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 $0$ .

$\frac{\frac{\log ({(x + 1)}^{5})}{\log (7)}}{\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} \log_{7} (x - 1)}$

Step 4.1.2

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

$\frac{\frac{\log ({(x + 1)}^{5})}{\log (7)}}{\frac{\log ({(x - 2)}^{3})}{\log (7)} \log_{7} (x - 1)}$

Step 4.2

Rewrite $\log_{7} (x - 1)$ using the change of base formula.

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Step 4.2.1

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 $0$ .

$\frac{\frac{\log ({(x + 1)}^{5})}{\log (7)}}{\frac{\log ({(x - 2)}^{3})}{\log (7)} \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}}$

Step 4.2.2

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

$\frac{\frac{\log ({(x + 1)}^{5})}{\log (7)}}{\frac{\log ({(x - 2)}^{3})}{\log (7)} \cdot \frac{\log (x - 1)}{\log (7)}}$

Step 5

Multiply $\frac{\log ({(x - 2)}^{3})}{\log (7)}$ by $\frac{\log (x - 1)}{\log (7)}$ .

$\frac{\frac{\log ({(x + 1)}^{5})}{\log (7)}}{\frac{\log ({(x - 2)}^{3}) \log (x - 1)}{\log (7) \log (7)}}$

Step 6

Simplify the denominator.

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Step 6.1

Raise $\log (7)$ to the power of $1$ .

$\frac{\frac{\log ({(x + 1)}^{5})}{\log (7)}}{\frac{\log ({(x - 2)}^{3}) \log (x - 1)}{\log^{1} (7) \log (7)}}$

Step 6.2

Raise $\log (7)$ to the power of $1$ .

$\frac{\frac{\log ({(x + 1)}^{5})}{\log (7)}}{\frac{\log ({(x - 2)}^{3}) \log (x - 1)}{\log^{1} (7) \log^{1} (7)}}$

Step 6.3

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

$\frac{\frac{\log ({(x + 1)}^{5})}{\log (7)}}{\frac{\log ({(x - 2)}^{3}) \log (x - 1)}{{\log (7)}^{1 + 1}}}$

Step 6.4

Add $1$ and $1$ .

$\frac{\frac{\log ({(x + 1)}^{5})}{\log (7)}}{\frac{\log ({(x - 2)}^{3}) \log (x - 1)}{\log^{2} (7)}}$

Step 7

Multiply the numerator by the reciprocal of the denominator.

$\frac{\log ({(x + 1)}^{5})}{\log (7)} \cdot \frac{\log^{2} (7)}{\log ({(x - 2)}^{3}) \log (x - 1)}$

Step 8

Combine.

$\frac{\log ({(x + 1)}^{5}) \log^{2} (7)}{\log (7) (\log ({(x - 2)}^{3}) \log (x - 1))}$

Step 9

Expand $\log ({(x + 1)}^{5})$ by moving $5$ outside the logarithm.

$\frac{5 \log (x + 1) \log^{2} (7)}{\log (7) (\log ({(x - 2)}^{3}) \log (x - 1))}$

Step 10

Expand $\log ({(x - 2)}^{3})$ by moving $3$ outside the logarithm.

$\frac{5 \log (x + 1) \log^{2} (7)}{\log (7) (3 \log (x - 2) \log (x - 1))}$

Step 11

Cancel the common factor of $\log^{2} (7)$ and $\log (7)$ .

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Step 11.1

Factor $\log (7)$ out of $5 \log (x + 1) \log^{2} (7)$ .

$\frac{\log (7) (5 \log (x + 1) \log (7))}{\log (7) (3 \log (x - 2) \log (x - 1))}$

Step 11.2

Cancel the common factors.

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Step 11.2.1

Cancel the common factor.

$\frac{\log (7) (5 \log (x + 1) \log (7))}{\log (7) (3 \log (x - 2) \log (x - 1))}$

Step 11.2.2

Rewrite the expression.

$\frac{5 \log (x + 1) \log (7)}{3 \log (x - 2) \log (x - 1)}$

Step 12

Simplify $5 \log (x + 1)$ by moving $5$ inside the logarithm.

$\frac{\log ({(x + 1)}^{5}) \log (7)}{3 \log (x - 2) \log (x - 1)}$

Step 13

Simplify $3 \log (x - 2)$ by moving $3$ inside the logarithm.

$\frac{\log ({(x + 1)}^{5}) \log (7)}{\log ({(x - 2)}^{3}) \log (x - 1)}$