Finite Math Examples

$\log_{b} (x) = \frac{1}{2} \cdot \log_{b} (4) + \frac{2}{3} \cdot \log_{b} (125) - \log_{b} (10)$

Step 1

Simplify the right side.

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Step 1.1

Simplify each term.

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Step 1.1.1

Combine $\frac{1}{2}$ and $\log_{b} (4)$ .

$\log_{b} (x) = \frac{\log_{b} (4)}{2} + \frac{2}{3} \log_{b} (125) - \log_{b} (10)$

Step 1.1.2

Combine $\frac{2}{3}$ and $\log_{b} (125)$ .

$\log_{b} (x) = \frac{\log_{b} (4)}{2} + \frac{2 \log_{b} (125)}{3} - \log_{b} (10)$

Step 2

Multiply each term in $\log_{b} (x) = \frac{\log_{b} (4)}{2} + \frac{2 \log_{b} (125)}{3} - \log_{b} (10)$ by $6$ to eliminate the fractions.

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Step 2.1

Multiply each term in $\log_{b} (x) = \frac{\log_{b} (4)}{2} + \frac{2 \log_{b} (125)}{3} - \log_{b} (10)$ by $6$ .

$\log_{b} (x) \cdot 6 = \frac{\log_{b} (4)}{2} \cdot 6 + \frac{2 \log_{b} (125)}{3} \cdot 6 - \log_{b} (10) \cdot 6$

Step 2.2

Simplify the left side.

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Step 2.2.1

Move $6$ to the left of $\log_{b} (x)$ .

$6 \log_{b} (x) = \frac{\log_{b} (4)}{2} \cdot 6 + \frac{2 \log_{b} (125)}{3} \cdot 6 - \log_{b} (10) \cdot 6$

Step 2.3

Simplify the right side.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Cancel the common factor of $2$ .

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Step 2.3.1.1.1

Factor $2$ out of $6$ .

$6 \log_{b} (x) = \frac{\log_{b} (4)}{2} \cdot (2 (3)) + \frac{2 \log_{b} (125)}{3} \cdot 6 - \log_{b} (10) \cdot 6$

Step 2.3.1.1.2

Cancel the common factor.

$6 \log_{b} (x) = \frac{\log_{b} (4)}{2} \cdot (2 \cdot 3) + \frac{2 \log_{b} (125)}{3} \cdot 6 - \log_{b} (10) \cdot 6$

Step 2.3.1.1.3

Rewrite the expression.

$6 \log_{b} (x) = \log_{b} (4) \cdot 3 + \frac{2 \log_{b} (125)}{3} \cdot 6 - \log_{b} (10) \cdot 6$

Step 2.3.1.2

Move $3$ to the left of $\log_{b} (4)$ .

$6 \log_{b} (x) = 3 \cdot \log_{b} (4) + \frac{2 \log_{b} (125)}{3} \cdot 6 - \log_{b} (10) \cdot 6$

Step 2.3.1.3

Cancel the common factor of $3$ .

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Step 2.3.1.3.1

Factor $3$ out of $6$ .

$6 \log_{b} (x) = 3 \log_{b} (4) + \frac{2 \log_{b} (125)}{3} \cdot (3 (2)) - \log_{b} (10) \cdot 6$

Step 2.3.1.3.2

Cancel the common factor.

$6 \log_{b} (x) = 3 \log_{b} (4) + \frac{2 \log_{b} (125)}{3} \cdot (3 \cdot 2) - \log_{b} (10) \cdot 6$

Step 2.3.1.3.3

Rewrite the expression.

$6 \log_{b} (x) = 3 \log_{b} (4) + 2 \log_{b} (125) \cdot 2 - \log_{b} (10) \cdot 6$

Step 2.3.1.4

Multiply $2$ by $2$ .

$6 \log_{b} (x) = 3 \log_{b} (4) + 4 \log_{b} (125) - \log_{b} (10) \cdot 6$

Step 2.3.1.5

Multiply $6$ by $- 1$ .

$6 \log_{b} (x) = 3 \log_{b} (4) + 4 \log_{b} (125) - 6 \log_{b} (10)$

Step 3

Simplify the left side.

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Step 3.1

Simplify $6 \log_{b} (x)$ by moving $6$ inside the logarithm.

$\log_{b} (x^{6}) = 3 \log_{b} (4) + 4 \log_{b} (125) - 6 \log_{b} (10)$

Step 4

Simplify the right side.

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Step 4.1

Simplify $3 \log_{b} (4) + 4 \log_{b} (125) - 6 \log_{b} (10)$ .

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Step 4.1.1

Simplify each term.

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Step 4.1.1.1

Simplify $3 \log_{b} (4)$ by moving $3$ inside the logarithm.

$\log_{b} (x^{6}) = \log_{b} (4^{3}) + 4 \log_{b} (125) - 6 \log_{b} (10)$

Step 4.1.1.2

Raise $4$ to the power of $3$ .

$\log_{b} (x^{6}) = \log_{b} (64) + 4 \log_{b} (125) - 6 \log_{b} (10)$

Step 4.1.1.3

Simplify $4 \log_{b} (125)$ by moving $4$ inside the logarithm.

$\log_{b} (x^{6}) = \log_{b} (64) + \log_{b} (125^{4}) - 6 \log_{b} (10)$

Step 4.1.1.4

Raise $125$ to the power of $4$ .

$\log_{b} (x^{6}) = \log_{b} (64) + \log_{b} (244140625) - 6 \log_{b} (10)$

Step 4.1.1.5

Simplify $- 6 \log_{b} (10)$ by moving $6$ inside the logarithm.

$\log_{b} (x^{6}) = \log_{b} (64) + \log_{b} (244140625) - \log_{b} (10^{6})$

Step 4.1.1.6

Raise $10$ to the power of $6$ .

$\log_{b} (x^{6}) = \log_{b} (64) + \log_{b} (244140625) - \log_{b} (1000000)$

Step 4.1.2

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

$\log_{b} (x^{6}) = \log_{b} (64 \cdot 244140625) - \log_{b} (1000000)$

Step 4.1.3

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

$\log_{b} (x^{6}) = \log_{b} (\frac{64 \cdot 244140625}{1000000})$

Step 4.1.4

Cancel the common factor of $64$ and $1000000$ .

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Step 4.1.4.1

Factor $64$ out of $64 \cdot 244140625$ .

$\log_{b} (x^{6}) = \log_{b} (\frac{64 (244140625)}{1000000})$

Step 4.1.4.2

Cancel the common factors.

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Step 4.1.4.2.1

Factor $64$ out of $1000000$ .

$\log_{b} (x^{6}) = \log_{b} (\frac{64 \cdot 244140625}{64 \cdot 15625})$

Step 4.1.4.2.2

Cancel the common factor.

$\log_{b} (x^{6}) = \log_{b} (\frac{64 \cdot 244140625}{64 \cdot 15625})$

Step 4.1.4.2.3

Rewrite the expression.

$\log_{b} (x^{6}) = \log_{b} (\frac{244140625}{15625})$

Step 4.1.5

Cancel the common factor of $244140625$ and $15625$ .

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Step 4.1.5.1

Factor $15625$ out of $244140625$ .

$\log_{b} (x^{6}) = \log_{b} (\frac{15625 \cdot 15625}{15625})$

Step 4.1.5.2

Cancel the common factors.

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Step 4.1.5.2.1

Factor $15625$ out of $15625$ .

$\log_{b} (x^{6}) = \log_{b} (\frac{15625 \cdot 15625}{15625 (1)})$

Step 4.1.5.2.2

Cancel the common factor.

$\log_{b} (x^{6}) = \log_{b} (\frac{15625 \cdot 15625}{15625 \cdot 1})$

Step 4.1.5.2.3

Rewrite the expression.

$\log_{b} (x^{6}) = \log_{b} (\frac{15625}{1})$

Step 4.1.5.2.4

Divide $15625$ by $1$ .

$\log_{b} (x^{6}) = \log_{b} (15625)$

Step 5

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

$x^{6} = 15625$

Step 6

Solve for $b$ .

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Step 6.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt[6]{15625}$

Step 6.2

Simplify $\pm \sqrt[6]{15625}$ .

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Step 6.2.1

Rewrite $15625$ as $5^{6}$ .

$x = \pm \sqrt[6]{5^{6}}$

Step 6.2.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 5$

Step 6.3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 6.3.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 5$

Step 6.3.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 5$

Step 6.3.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 5, - 5$

Step 7

The variable $b$ got canceled.

All real numbers

Step 8

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$