Trigonometry Examples

$6 \log_{4} (4 x + 5) + 2 = 5$

Step 1

Move all terms not containing $\log_{4} (4 x + 5)$ to the right side of the equation.

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Step 1.1

Subtract $2$ from both sides of the equation.

$6 \log_{4} (4 x + 5) = 5 - 2$

Step 1.2

Subtract $2$ from $5$ .

$6 \log_{4} (4 x + 5) = 3$

Step 2

Divide each term in $6 \log_{4} (4 x + 5) = 3$ by $6$ and simplify.

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Step 2.1

Divide each term in $6 \log_{4} (4 x + 5) = 3$ by $6$ .

$\frac{6 \log_{4} (4 x + 5)}{6} = \frac{3}{6}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $6$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{6 \log_{4} (4 x + 5)}{6} = \frac{3}{6}$

Step 2.2.1.2

Divide $\log_{4} (4 x + 5)$ by $1$ .

$\log_{4} (4 x + 5) = \frac{3}{6}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Cancel the common factor of $3$ and $6$ .

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Step 2.3.1.1

Factor $3$ out of $3$ .

$\log_{4} (4 x + 5) = \frac{3 (1)}{6}$

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

Factor $3$ out of $6$ .

$\log_{4} (4 x + 5) = \frac{3 \cdot 1}{3 \cdot 2}$

Step 2.3.1.2.2

Cancel the common factor.

$\log_{4} (4 x + 5) = \frac{3 \cdot 1}{3 \cdot 2}$

Step 2.3.1.2.3

Rewrite the expression.

$\log_{4} (4 x + 5) = \frac{1}{2}$

Step 3

Rewrite $\log_{4} (4 x + 5) = \frac{1}{2}$ 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 $b^{y} = x$ .

$4^{\frac{1}{2}} = 4 x + 5$

Step 4

Solve for $x$ .

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Step 4.1

Rewrite the equation as $4 x + 5 = 4^{\frac{1}{2}}$ .

$4 x + 5 = 4^{\frac{1}{2}}$

Step 4.2

Simplify $4^{\frac{1}{2}}$ .

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Step 4.2.1

Rewrite.

$4 x + 5 = 0 + 0 + 4^{\frac{1}{2}}$

Step 4.2.2

Simplify the expression.

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Step 4.2.2.1

Rewrite $4$ as $2^{2}$ .

$4 x + 5 = {(2^{2})}^{\frac{1}{2}}$

Step 4.2.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$4 x + 5 = 2^{2 (\frac{1}{2})}$

Step 4.2.3

Cancel the common factor of $2$ .

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Step 4.2.3.1

Cancel the common factor.

$4 x + 5 = 2^{2 (\frac{1}{2})}$

Step 4.2.3.2

Rewrite the expression.

$4 x + 5 = 2^{1}$

Step 4.2.4

Evaluate the exponent.

$4 x + 5 = 2$

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 $5$ from both sides of the equation.

$4 x = 2 - 5$

Step 4.3.2

Subtract $5$ from $2$ .

$4 x = - 3$

Step 4.4

Divide each term in $4 x = - 3$ by $4$ and simplify.

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Step 4.4.1

Divide each term in $4 x = - 3$ by $4$ .

$\frac{4 x}{4} = \frac{- 3}{4}$

Step 4.4.2

Simplify the left side.

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Step 4.4.2.1

Cancel the common factor of $4$ .

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Step 4.4.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 3}{4}$

Step 4.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 3}{4}$

Step 4.4.3

Simplify the right side.

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Step 4.4.3.1

Move the negative in front of the fraction.

$x = - \frac{3}{4}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{3}{4}$

Decimal Form:

$x = - 0.75$