Precalculus Examples

$2 \log (x) = \log (2) + \log (5 x - 8)$

Step 1

Simplify the left side.

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Step 1.1

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

$\log (x^{2}) = \log (2) + \log (5 x - 8)$

Step 2

Simplify the right side.

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Step 2.1

Simplify $\log (2) + \log (5 x - 8)$ .

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Step 2.1.1

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

$\log (x^{2}) = \log (2 (5 x - 8))$

Step 2.1.2

Apply the distributive property.

$\log (x^{2}) = \log (2 (5 x) + 2 \cdot - 8)$

Step 2.1.3

Multiply.

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Step 2.1.3.1

Multiply $5$ by $2$ .

$\log (x^{2}) = \log (10 x + 2 \cdot - 8)$

Step 2.1.3.2

Multiply $2$ by $- 8$ .

$\log (x^{2}) = \log (10 x - 16)$

Step 3

For the equation to be equal, the argument of the logarithms on both sides of the equation must be equal.

$x^{2} = 10 x - 16$

Step 4

Solve for $x$ .

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Step 4.1

Subtract $10 x$ from both sides of the equation.

$x^{2} - 10 x = - 16$

Step 4.2

Add $16$ to both sides of the equation.

$x^{2} - 10 x + 16 = 0$

Step 4.3

Factor $x^{2} - 10 x + 16$ using the AC method.

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Step 4.3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $16$ and whose sum is $- 10$ .

$- 8, - 2$

Step 4.3.2

Write the factored form using these integers.

$(x - 8) (x - 2) = 0$

Step 4.4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 8 = 0$

$x - 2 = 0$

Step 4.5

Set $x - 8$ equal to $0$ and solve for $x$ .

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Step 4.5.1

Set $x - 8$ equal to $0$ .

$x - 8 = 0$

Step 4.5.2

Add $8$ to both sides of the equation.

$x = 8$

Step 4.6

Set $x - 2$ equal to $0$ and solve for $x$ .

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Step 4.6.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 4.6.2

Add $2$ to both sides of the equation.

$x = 2$

Step 4.7

The final solution is all the values that make $(x - 8) (x - 2) = 0$ true.

$x = 8, 2$