Trigonometry Examples

$\log_{3} (3 x + 5) - 2 \log_{3} (3) = 4$

Step 1

Simplify each term.

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Step 1.1

Logarithm base $3$ of $3$ is $1$ .

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

Step 1.2

Multiply $- 2$ by $1$ .

$\log_{3} (3 x + 5) - 2 = 4$

Step 2

Move all terms not containing $\log_{3} (3 x + 5)$ to the right side of the equation.

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Step 2.1

Add $2$ to both sides of the equation.

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

Step 2.2

Add $4$ and $2$ .

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

Step 3

Rewrite $\log_{3} (3 x + 5) = 6$ 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 $b^{y} = x$ .

$3^{6} = 3 x + 5$

Step 4

Solve for $x$ .

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Step 4.1

Rewrite the equation as $3 x + 5 = 3^{6}$ .

$3 x + 5 = 3^{6}$

Step 4.2

Raise $3$ to the power of $6$ .

$3 x + 5 = 729$

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.

$3 x = 729 - 5$

Step 4.3.2

Subtract $5$ from $729$ .

$3 x = 724$

Step 4.4

Divide each term in $3 x = 724$ by $3$ and simplify.

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Step 4.4.1

Divide each term in $3 x = 724$ by $3$ .

$\frac{3 x}{3} = \frac{724}{3}$

Step 4.4.2

Simplify the left side.

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Step 4.4.2.1

Cancel the common factor of $3$ .

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Step 4.4.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{724}{3}$

Step 4.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{724}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{724}{3}$

Decimal Form:

$x = 241.\overline{3}$

Mixed Number Form:

$x = 241 \frac{1}{3}$