Finite Math Examples

$\log_{b} (x) = \frac{2}{3} \cdot (\log_{b} (64 + \frac{1}{2}) \log_{b} (25)) - \log_{b} (20)$

Step 1

Simplify the right side.

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Step 1.1

Simplify each term.

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Step 1.1.1

Multiply $\frac{2}{3} (\log_{b} (64 + \frac{1}{2}) \log_{b} (25))$ .

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Step 1.1.1.1

Combine $\log_{b} (64 + \frac{1}{2})$ and $\frac{2}{3}$ .

$\log_{b} (x) = \frac{\log_{b} (64 + \frac{1}{2}) \cdot 2}{3} \log_{b} (25) - \log_{b} (20)$

Step 1.1.1.2

Combine $\frac{\log_{b} (64 + \frac{1}{2}) \cdot 2}{3}$ and $\log_{b} (25)$ .

$\log_{b} (x) = \frac{\log_{b} (64 + \frac{1}{2}) \cdot 2 \log_{b} (25)}{3} - \log_{b} (20)$

Step 1.1.2

Move $2$ to the left of $\log_{b} (64 + \frac{1}{2})$ .

$\log_{b} (x) = \frac{2 \log_{b} (64 + \frac{1}{2}) \log_{b} (25)}{3} - \log_{b} (20)$

Step 2

Simplify the right side.

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Step 2.1

Simplify $\frac{2 \log_{b} (64 + \frac{1}{2}) \log_{b} (25)}{3} - \log_{b} (20)$ .

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Step 2.1.1

To write $64$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\log_{b} (x) = \frac{2 \log_{b} (64 \cdot \frac{2}{2} + \frac{1}{2}) \log_{b} (25)}{3} - \log_{b} (20)$

Step 2.1.2

Combine $64$ and $\frac{2}{2}$ .

$\log_{b} (x) = \frac{2 \log_{b} (\frac{64 \cdot 2}{2} + \frac{1}{2}) \log_{b} (25)}{3} - \log_{b} (20)$

Step 2.1.3

Combine the numerators over the common denominator.

$\log_{b} (x) = \frac{2 \log_{b} (\frac{64 \cdot 2 + 1}{2}) \log_{b} (25)}{3} - \log_{b} (20)$

Step 2.1.4

Simplify the numerator.

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Step 2.1.4.1

Multiply $64$ by $2$ .

$\log_{b} (x) = \frac{2 \log_{b} (\frac{128 + 1}{2}) \log_{b} (25)}{3} - \log_{b} (20)$

Step 2.1.4.2

Add $128$ and $1$ .

$\log_{b} (x) = \frac{2 \log_{b} (\frac{129}{2}) \log_{b} (25)}{3} - \log_{b} (20)$

Step 2.1.5

Simplify each term.

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Step 2.1.5.1

Simplify $2 \log_{b} (\frac{129}{2})$ by moving $2$ inside the logarithm.

$\log_{b} (x) = \frac{\log_{b} ({(\frac{129}{2})}^{2}) \log_{b} (25)}{3} - \log_{b} (20)$

Step 2.1.5.2

Simplify the numerator.

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Step 2.1.5.2.1

Apply the product rule to $\frac{129}{2}$ .

$\log_{b} (x) = \frac{\log_{b} (\frac{129^{2}}{2^{2}}) \log_{b} (25)}{3} - \log_{b} (20)$

Step 2.1.5.2.2

Raise $129$ to the power of $2$ .

$\log_{b} (x) = \frac{\log_{b} (\frac{16641}{2^{2}}) \log_{b} (25)}{3} - \log_{b} (20)$

Step 2.1.5.2.3

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

$\log_{b} (x) = \frac{\log_{b} (\frac{16641}{4}) \log_{b} (25)}{3} - \log_{b} (20)$

Step 3

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

$\log_{b} (x) - \frac{\log_{b} (\frac{16641}{4}) \log_{b} (25)}{3} + \log_{b} (20) = 0$

Step 4

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

$\log_{b} (x \cdot 20) - \frac{\log_{b} (\frac{16641}{4}) \log_{b} (25)}{3} = 0$

Step 5

Move $20$ to the left of $x$ .

$\log_{b} (20 x) - \frac{\log_{b} (\frac{16641}{4}) \log_{b} (25)}{3} = 0$

Step 6

Add $\frac{\log_{b} (\frac{16641}{4}) \log_{b} (25)}{3}$ to both sides of the equation.

$\log_{b} (20 x) = \frac{\log_{b} (\frac{16641}{4}) \log_{b} (25)}{3}$

Step 7

Rewrite $\log_{b} (20 x) = \frac{\log_{b} (\frac{16641}{4}) \log_{b} (25)}{3}$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$b^{\frac{\log_{b} (\frac{16641}{4}) \log_{b} (25)}{3}} = 20 x$

Step 8

Solve for $x$ .

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Step 8.1

Rewrite the equation as $20 x = b^{\frac{\log_{b} (\frac{16641}{4}) \log_{b} (25)}{3}}$ .

$20 x = b^{\frac{\log_{b} (\frac{16641}{4}) \log_{b} (25)}{3}}$

Step 8.2

Divide each term in $20 x = b^{\frac{\log_{b} (\frac{16641}{4}) \log_{b} (25)}{3}}$ by $20$ and simplify.

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Step 8.2.1

Divide each term in $20 x = b^{\frac{\log_{b} (\frac{16641}{4}) \log_{b} (25)}{3}}$ by $20$ .

$\frac{20 x}{20} = \frac{b^{\frac{\log_{b} (\frac{16641}{4}) \log_{b} (25)}{3}}}{20}$

Step 8.2.2

Simplify the left side.

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Step 8.2.2.1

Cancel the common factor of $20$ .

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Step 8.2.2.1.1

Cancel the common factor.

$\frac{20 x}{20} = \frac{b^{\frac{\log_{b} (\frac{16641}{4}) \log_{b} (25)}{3}}}{20}$

Step 8.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{b^{\frac{\log_{b} (\frac{16641}{4}) \log_{b} (25)}{3}}}{20}$