Finite Math Examples

$\log ({(b)}^{x}) + \log (b) (x - 4) = \log (b) \cdot 21$

Step 1

Simplify the left side.

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

$\log (b^{x}) + \log (b) x + \log (b) \cdot - 4 = \log (b) \cdot 21$

Step 1.1.2

Move $- 4$ to the left of $\log (b)$ .

$\log (b^{x}) + \log (b) x - 4 \log (b) = \log (b) \cdot 21$

Step 1.2

Simplify the expression.

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Step 1.2.1

Reorder factors in $\log (b^{x}) + \log (b) x - 4 \log (b)$ .

$\log (b^{x}) + x \log (b) - 4 \log (b) = \log (b) \cdot 21$

Step 1.2.2

Reorder $\log (b^{x})$ and $x \log (b)$ .

$x \log (b) + \log (b^{x}) - 4 \log (b) = \log (b) \cdot 21$

Step 2

Move $21$ to the left of $\log (b)$ .

$x \log (b) + \log (b^{x}) - 4 \log (b) = 21 \log (b)$

Step 3

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

$x \log (b) + \log (b^{x}) - 4 \log (b) - 21 \log (b) = 0$

Step 4

Subtract $21 \log (b)$ from $- 4 \log (b)$ .

$x \log (b) + \log (b^{x}) - 25 \log (b) = 0$

Step 5

Simplify the left side.

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Step 5.1

Simplify $x \log (b) + \log (b^{x}) - 25 \log (b)$ .

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Step 5.1.1

Simplify $- 25 \log (b)$ by moving $25$ inside the logarithm.

$x \log (b) + \log (b^{x}) - \log (b^{25}) = 0$

Step 5.1.2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$x \log (b) + \log (\frac{b^{x}}{b^{25}}) = 0$

Step 5.1.3

Cancel the common factor of $b^{x}$ and $b^{25}$ .

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Step 5.1.3.1

Factor $b^{25}$ out of $b^{x}$ .

$x \log (b) + \log (\frac{b^{25} b^{x - 25}}{b^{25}}) = 0$

Step 5.1.3.2

Cancel the common factors.

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Step 5.1.3.2.1

Multiply by $1$ .

$x \log (b) + \log (\frac{b^{25} b^{x - 25}}{b^{25} \cdot 1}) = 0$

Step 5.1.3.2.2

Cancel the common factor.

$x \log (b) + \log (\frac{b^{25} b^{x - 25}}{b^{25} \cdot 1}) = 0$

Step 5.1.3.2.3

Rewrite the expression.

$x \log (b) + \log (\frac{b^{x - 25}}{1}) = 0$

Step 5.1.3.2.4

Divide $b^{x - 25}$ by $1$ .

$x \log (b) + \log (b^{x - 25}) = 0$