Precalculus Examples

$3 \log (5) (x + 5) - \log (5) (x - 19) - \log (5) (x - 8)$

Step 1

Simplify each term.

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Step 1.1

Simplify $3 \log (5)$ by moving $3$ inside the logarithm.

$\log (5^{3}) (x + 5) - \log (5) (x - 19) - \log (5) (x - 8)$

Step 1.2

Raise $5$ to the power of $3$ .

$\log (125) (x + 5) - \log (5) (x - 19) - \log (5) (x - 8)$

Step 1.3

Apply the distributive property.

$\log (125) x + \log (125) \cdot 5 - \log (5) (x - 19) - \log (5) (x - 8)$

Step 1.4

Multiply $\log (125) \cdot 5$ .

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Step 1.4.1

Reorder $\log (125)$ and $5$ .

$\log (125) x + 5 \cdot \log (125) - \log (5) (x - 19) - \log (5) (x - 8)$

Step 1.4.2

Simplify $5 \cdot \log (125)$ by moving $5$ inside the logarithm.

$\log (125) x + \log (125^{5}) - \log (5) (x - 19) - \log (5) (x - 8)$

Step 1.5

Raise $125$ to the power of $5$ .

$\log (125) x + \log (30517578125) - \log (5) (x - 19) - \log (5) (x - 8)$

Step 1.6

Apply the distributive property.

$\log (125) x + \log (30517578125) - \log (5) x - \log (5) \cdot - 19 - \log (5) (x - 8)$

Step 1.7

Multiply $- \log (5) \cdot - 19$ .

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Step 1.7.1

Multiply $- 19$ by $- 1$ .

$\log (125) x + \log (30517578125) - \log (5) x + 19 \log (5) - \log (5) (x - 8)$

Step 1.7.2

Simplify $19 \log (5)$ by moving $19$ inside the logarithm.

$\log (125) x + \log (30517578125) - \log (5) x + \log (5^{19}) - \log (5) (x - 8)$

Step 1.8

Raise $5$ to the power of $19$ .

$\log (125) x + \log (30517578125) - \log (5) x + \log (19073486328125) - \log (5) (x - 8)$

Step 1.9

Apply the distributive property.

$\log (125) x + \log (30517578125) - \log (5) x + \log (19073486328125) - \log (5) x - \log (5) \cdot - 8$

Step 1.10

Multiply $- \log (5) \cdot - 8$ .

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Step 1.10.1

Multiply $- 8$ by $- 1$ .

$\log (125) x + \log (30517578125) - \log (5) x + \log (19073486328125) - \log (5) x + 8 \log (5)$

Step 1.10.2

Simplify $8 \log (5)$ by moving $8$ inside the logarithm.

$\log (125) x + \log (30517578125) - \log (5) x + \log (19073486328125) - \log (5) x + \log (5^{8})$

Step 1.11

Raise $5$ to the power of $8$ .

$\log (125) x + \log (30517578125) - \log (5) x + \log (19073486328125) - \log (5) x + \log (390625)$

Step 2

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\log (125) x - \log (5) x + \log (30517578125 \cdot 19073486328125) - \log (5) x + \log (390625)$

Step 3

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\log (125) x - \log (5) x - \log (5) x + \log (30517578125 \cdot 19073486328125 \cdot 390625)$

Step 4

Simplify by adding terms.

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Step 4.1

Multiply $30517578125$ by $19073486328125$ .

$\log (125) x - \log (5) x - \log (5) x + \log (582076609134674072265625 \cdot 390625)$

Step 4.2

Subtract $\log (5) x$ from $- \log (5) x$ .

$\log (125) x - 2 \log (5) x + \log (582076609134674072265625 \cdot 390625)$

Step 5

Simplify each term.

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Step 5.1

Simplify $- 2 \log (5)$ by moving $2$ inside the logarithm.

$\log (125) x - \log (5^{2}) x + \log (582076609134674072265625 \cdot 390625)$

Step 5.2

Raise $5$ to the power of $2$ .

$\log (125) x - \log (25) x + \log (582076609134674072265625 \cdot 390625)$

Step 6

Reorder factors in $\log (125) x - \log (25) x + \log (582076609134674072265625 \cdot 390625)$ .

$x \log (125) - x \log (25) + \log (582076609134674072265625 \cdot 390625)$