Calculus Examples

$8 \log_{4} (2) + \log_{4} (3 + x) = 1$

Step 1

Simplify the left side.

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Step 1.1

Simplify each term.

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Step 1.1.1

Logarithm base $4$ of $2$ is $\frac{1}{2}$ .

$8 (\frac{1}{2}) + \log_{4} (3 + x) = 1$

Step 1.1.2

Cancel the common factor of $2$ .

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Step 1.1.2.1

Factor $2$ out of $8$ .

$2 (4) (\frac{1}{2}) + \log_{4} (3 + x) = 1$

Step 1.1.2.2

Cancel the common factor.

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

Step 1.1.2.3

Rewrite the expression.

$4 + \log_{4} (3 + x) = 1$

Step 1.2

Reorder $3$ and $x$ .

$4 + \log_{4} (x + 3) = 1$

Step 2

Move all terms not containing $\log_{4} (x + 3)$ to the right side of the equation.

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Step 2.1

Subtract $4$ from both sides of the equation.

$\log_{4} (x + 3) = 1 - 4$

Step 2.2

Subtract $4$ from $1$ .

$\log_{4} (x + 3) = - 3$

Step 3

Rewrite $\log_{4} (x + 3) = - 3$ 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 $b^{y} = x$ .

$4^{- 3} = x + 3$

Step 4

Solve for $x$ .

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Step 4.1

Rewrite the equation as $x + 3 = 4^{- 3}$ .

$x + 3 = 4^{- 3}$

Step 4.2

Simplify $4^{- 3}$ .

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Step 4.2.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$x + 3 = \frac{1}{4^{3}}$

Step 4.2.2

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

$x + 3 = \frac{1}{64}$

Step 4.3

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

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Step 4.3.1

Subtract $3$ from both sides of the equation.

$x = \frac{1}{64} - 3$

Step 4.3.2

To write $- 3$ as a fraction with a common denominator, multiply by $\frac{64}{64}$ .

$x = \frac{1}{64} - 3 \cdot \frac{64}{64}$

Step 4.3.3

Combine $- 3$ and $\frac{64}{64}$ .

$x = \frac{1}{64} + \frac{- 3 \cdot 64}{64}$

Step 4.3.4

Combine the numerators over the common denominator.

$x = \frac{1 - 3 \cdot 64}{64}$

Step 4.3.5

Simplify the numerator.

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Step 4.3.5.1

Multiply $- 3$ by $64$ .

$x = \frac{1 - 192}{64}$

Step 4.3.5.2

Subtract $192$ from $1$ .

$x = \frac{- 191}{64}$

Step 4.3.6

Move the negative in front of the fraction.

$x = - \frac{191}{64}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{191}{64}$

Decimal Form:

$x = - 2.984375$

Mixed Number Form:

$x = - 2 \frac{63}{64}$