Precalculus Examples

$(2 x - 1) \cdot (5 \log (2)) = (x - 1) \cdot (7 \log (3))$

Step 1

Simplify the expressions in the equation.

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Step 1.1

Simplify the left side.

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Step 1.1.1

Simplify $(2 x - 1) \cdot (5 \log (2))$ .

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Step 1.1.1.1

Apply the distributive property.

$2 x (5 \log (2)) - 1 (5 \log (2)) = (x - 1) \cdot (7 \log (3))$

Step 1.1.1.2

Simplify the expression.

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Step 1.1.1.2.1

Rewrite using the commutative property of multiplication.

$2 \cdot 5 x \log (2) - 1 (5 \log (2)) = (x - 1) \cdot (7 \log (3))$

Step 1.1.1.2.2

Multiply $5$ by $- 1$ .

$2 \cdot 5 x \log (2) - 5 \log (2) = (x - 1) \cdot (7 \log (3))$

Step 1.1.1.2.3

Multiply $2$ by $5$ .

$10 x \log (2) - 5 \log (2) = (x - 1) \cdot (7 \log (3))$

Step 1.2

Simplify the right side.

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Step 1.2.1

Simplify $(x - 1) \cdot (7 \log (3))$ .

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Step 1.2.1.1

Apply the distributive property.

$10 x \log (2) - 5 \log (2) = x (7 \log (3)) - 1 (7 \log (3))$

Step 1.2.1.2

Simplify the expression.

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Step 1.2.1.2.1

Rewrite using the commutative property of multiplication.

$10 x \log (2) - 5 \log (2) = 7 x \log (3) - 1 (7 \log (3))$

Step 1.2.1.2.2

Multiply $7$ by $- 1$ .

$10 x \log (2) - 5 \log (2) = 7 x \log (3) - 7 \log (3)$

Step 2

Simplify the left side.

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Step 2.1

Simplify each term.

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Step 2.1.1

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

$x \log (2^{10}) - 5 \log (2) = 7 x \log (3) - 7 \log (3)$

Step 2.1.2

Raise $2$ to the power of $10$ .

$x \log (1024) - 5 \log (2) = 7 x \log (3) - 7 \log (3)$

Step 2.1.3

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

$x \log (1024) - \log (2^{5}) = 7 x \log (3) - 7 \log (3)$

Step 2.1.4

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

$x \log (1024) - \log (32) = 7 x \log (3) - 7 \log (3)$

Step 3

Simplify the right side.

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Step 3.1

Simplify each term.

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Step 3.1.1

Simplify $7 \log (3)$ by moving $7$ inside the logarithm.

$x \log (1024) - \log (32) = x \log (3^{7}) - 7 \log (3)$

Step 3.1.2

Raise $3$ to the power of $7$ .

$x \log (1024) - \log (32) = x \log (2187) - 7 \log (3)$

Step 3.1.3

Simplify $- 7 \log (3)$ by moving $7$ inside the logarithm.

$x \log (1024) - \log (32) = x \log (2187) - \log (3^{7})$

Step 3.1.4

Raise $3$ to the power of $7$ .

$x \log (1024) - \log (32) = x \log (2187) - \log (2187)$

Step 4

Move all the terms containing a logarithm to the left side of the equation.

$x \log (1024) - \log (32) - x \log (2187) + \log (2187) = 0$

Step 5

Move all terms not containing $x$ to the right side of the equation.

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Step 5.1

Add $\log (32)$ to both sides of the equation.

$x \log (1024) - x \log (2187) + \log (2187) = \log (32)$

Step 5.2

Subtract $\log (2187)$ from both sides of the equation.

$x \log (1024) - x \log (2187) = \log (32) - \log (2187)$

Step 6

Factor $x$ out of $x \log (1024) - x \log (2187)$ .

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Step 6.1

Factor $x$ out of $x \log (1024)$ .

$x (\log (1024)) - x \log (2187) = \log (32) - \log (2187)$

Step 6.2

Factor $x$ out of $- x \log (2187)$ .

$x (\log (1024)) + x (- 1 \log (2187)) = \log (32) - \log (2187)$

Step 6.3

Factor $x$ out of $x (\log (1024)) + x (- 1 \log (2187))$ .

$x (\log (1024) - 1 \log (2187)) = \log (32) - \log (2187)$

Step 7

Rewrite $- 1 \log (2187)$ as $- \log (2187)$ .

$x (\log (1024) - \log (2187)) = \log (32) - \log (2187)$

Step 8

Divide each term in $x (\log (1024) - \log (2187)) = \log (32) - \log (2187)$ by $\log (1024) - \log (2187)$ and simplify.

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Step 8.1

Divide each term in $x (\log (1024) - \log (2187)) = \log (32) - \log (2187)$ by $\log (1024) - \log (2187)$ .

$\frac{x (\log (1024) - \log (2187))}{\log (1024) - \log (2187)} = \frac{\log (32)}{\log (1024) - \log (2187)} + \frac{- \log (2187)}{\log (1024) - \log (2187)}$

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $\log (1024) - \log (2187)$ .

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Step 8.2.1.1

Cancel the common factor.

$\frac{x (\log (1024) - \log (2187))}{\log (1024) - \log (2187)} = \frac{\log (32)}{\log (1024) - \log (2187)} + \frac{- \log (2187)}{\log (1024) - \log (2187)}$

Step 8.2.1.2

Divide $x$ by $1$ .

$x = \frac{\log (32)}{\log (1024) - \log (2187)} + \frac{- \log (2187)}{\log (1024) - \log (2187)}$

Step 8.3

Simplify the right side.

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Step 8.3.1

Combine the numerators over the common denominator.

$x = \frac{\log (32) - \log (2187)}{\log (1024) - \log (2187)}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\log (32) - \log (2187)}{\log (1024) - \log (2187)}$

Decimal Form:

$x = 5.56730492 \dots$