Precalculus Examples

$\log (x) = \frac{1}{2} \cdot \log (16) - \frac{1}{3} \cdot \log (8) + 1$

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 $\log (16)$ .

$\log (x) = \frac{\log (16)}{2} - \frac{1}{3} \log (8) + 1$

Step 1.1.2

Combine $\log (8)$ and $\frac{1}{3}$ .

$\log (x) = \frac{\log (16)}{2} - \frac{\log (8)}{3} + 1$

Step 2

Multiply each term in $\log (x) = \frac{\log (16)}{2} - \frac{\log (8)}{3} + 1$ by $6$ to eliminate the fractions.

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Step 2.1

Multiply each term in $\log (x) = \frac{\log (16)}{2} - \frac{\log (8)}{3} + 1$ by $6$ .

$\log (x) \cdot 6 = \frac{\log (16)}{2} \cdot 6 - \frac{\log (8)}{3} \cdot 6 + 1 \cdot 6$

Step 2.2

Simplify the left side.

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Step 2.2.1

Move $6$ to the left of $\log (x)$ .

$6 \log (x) = \frac{\log (16)}{2} \cdot 6 - \frac{\log (8)}{3} \cdot 6 + 1 \cdot 6$

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 $6$ .

$6 \log (x) = \frac{\log (16)}{2} \cdot (2 (3)) - \frac{\log (8)}{3} \cdot 6 + 1 \cdot 6$

Step 2.3.1.1.2

Cancel the common factor.

$6 \log (x) = \frac{\log (16)}{2} \cdot (2 \cdot 3) - \frac{\log (8)}{3} \cdot 6 + 1 \cdot 6$

Step 2.3.1.1.3

Rewrite the expression.

$6 \log (x) = \log (16) \cdot 3 - \frac{\log (8)}{3} \cdot 6 + 1 \cdot 6$

Step 2.3.1.2

Move $3$ to the left of $\log (16)$ .

$6 \log (x) = 3 \cdot \log (16) - \frac{\log (8)}{3} \cdot 6 + 1 \cdot 6$

Step 2.3.1.3

Cancel the common factor of $3$ .

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Step 2.3.1.3.1

Move the leading negative in $- \frac{\log (8)}{3}$ into the numerator.

$6 \log (x) = 3 \log (16) + \frac{- \log (8)}{3} \cdot 6 + 1 \cdot 6$

Step 2.3.1.3.2

Factor $3$ out of $6$ .

$6 \log (x) = 3 \log (16) + \frac{- \log (8)}{3} \cdot (3 (2)) + 1 \cdot 6$

Step 2.3.1.3.3

Cancel the common factor.

$6 \log (x) = 3 \log (16) + \frac{- \log (8)}{3} \cdot (3 \cdot 2) + 1 \cdot 6$

Step 2.3.1.3.4

Rewrite the expression.

$6 \log (x) = 3 \log (16) - \log (8) \cdot 2 + 1 \cdot 6$

Step 2.3.1.4

Multiply $2$ by $- 1$ .

$6 \log (x) = 3 \log (16) - 2 \log (8) + 1 \cdot 6$

Step 2.3.1.5

Multiply $6$ by $1$ .

$6 \log (x) = 3 \log (16) - 2 \log (8) + 6$

Step 3

Simplify the left side.

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Step 3.1

Simplify $6 \log (x)$ by moving $6$ inside the logarithm.

$\log (x^{6}) = 3 \log (16) - 2 \log (8) + 6$

Step 4

Simplify the right side.

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Step 4.1

Simplify $3 \log (16) - 2 \log (8) + 6$ .

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Step 4.1.1

Simplify each term.

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Step 4.1.1.1

Simplify $3 \log (16)$ by moving $3$ inside the logarithm.

$\log (x^{6}) = \log (16^{3}) - 2 \log (8) + 6$

Step 4.1.1.2

Raise $16$ to the power of $3$ .

$\log (x^{6}) = \log (4096) - 2 \log (8) + 6$

Step 4.1.1.3

Simplify $- 2 \log (8)$ by moving $2$ inside the logarithm.

$\log (x^{6}) = \log (4096) - \log (8^{2}) + 6$

Step 4.1.1.4

Raise $8$ to the power of $2$ .

$\log (x^{6}) = \log (4096) - \log (64) + 6$

Step 4.1.2

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

$\log (x^{6}) = \log (\frac{4096}{64}) + 6$

Step 4.1.3

Divide $4096$ by $64$ .

$\log (x^{6}) = \log (64) + 6$

Step 5

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

$\log (x^{6}) - \log (64) = 6$

Step 6

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

$\log (\frac{x^{6}}{64}) = 6$

Step 7

Rewrite $\log (\frac{x^{6}}{64}) = 6$ 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 $b^{y} = x$ .

$10^{6} = \frac{x^{6}}{64}$

Step 8

Solve for $x$ .

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Step 8.1

Rewrite the equation as $\frac{x^{6}}{64} = 10^{6}$ .

$\frac{x^{6}}{64} = 10^{6}$

Step 8.2

Multiply both sides of the equation by $64$ .

$64 \frac{x^{6}}{64} = 64 \cdot 10^{6}$

Step 8.3

Simplify both sides of the equation.

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Step 8.3.1

Simplify the left side.

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Step 8.3.1.1

Cancel the common factor of $64$ .

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Step 8.3.1.1.1

Cancel the common factor.

$64 \frac{x^{6}}{64} = 64 \cdot 10^{6}$

Step 8.3.1.1.2

Rewrite the expression.

$x^{6} = 64 \cdot 10^{6}$

Step 8.3.2

Simplify the right side.

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Step 8.3.2.1

Move the decimal point in $64$ to the left by $1$ place and increase the power of $10^{6}$ by $1$ .

$x^{6} = 6.4 \cdot 10^{7}$

Step 8.4

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

$x = \pm \sqrt[6]{6.4 \cdot 10^{7}}$

Step 8.5

Simplify $\pm \sqrt[6]{6.4 \cdot 10^{7}}$ .

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Step 8.5.1

Rewrite $6.4 \cdot 10^{7}$ as $64 \cdot 10^{6}$ .

$x = \pm \sqrt[6]{64 \cdot 10^{6}}$

Step 8.5.2

Rewrite $\sqrt[6]{64 \cdot 10^{6}}$ as $\sqrt[6]{64} \cdot \sqrt[6]{10^{6}}$ .

$x = \pm \sqrt[6]{64} \cdot \sqrt[6]{10^{6}}$

Step 8.5.3

Evaluate the root.

$x = \pm 2 \cdot \sqrt[6]{10^{6}}$

Step 8.5.4

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

$x = \pm 2 \cdot 10$

Step 8.5.5

Raise $10$ to the power of $1$ .

$x = \pm 2 \cdot 10^{1}$

Step 8.6

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

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Step 8.6.1

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

$x = 2 \cdot 10^{1}$

Step 8.6.2

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

$x = - 2 \cdot 10^{1}$

Step 8.6.3

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

$x = 2 \cdot 10^{1}, - 2 \cdot 10^{1}$

Step 9

Exclude the solutions that do not make $\log (x) = \frac{1}{2} \cdot \log (16) - \frac{1}{3} \cdot \log (8) + 1$ true.

$x = 2 \cdot 10^{1}$

Step 10

The result can be shown in multiple forms.

Scientific Notation:

$x = 2 \cdot 10^{1}$

Expanded Form:

$x = 20$