Trigonometry Examples

$6 \ln (8 x + 9) - 6 = 12$

Step 1

Move all terms not containing $\ln (8 x + 9)$ to the right side of the equation.

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Step 1.1

Add $6$ to both sides of the equation.

$6 \ln (8 x + 9) = 12 + 6$

Step 1.2

Add $12$ and $6$ .

$6 \ln (8 x + 9) = 18$

Step 2

Divide each term in $6 \ln (8 x + 9) = 18$ by $6$ and simplify.

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Step 2.1

Divide each term in $6 \ln (8 x + 9) = 18$ by $6$ .

$\frac{6 \ln (8 x + 9)}{6} = \frac{18}{6}$

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.

$\frac{6 \ln (8 x + 9)}{6} = \frac{18}{6}$

Step 2.2.1.2

Divide $\ln (8 x + 9)$ by $1$ .

$\ln (8 x + 9) = \frac{18}{6}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $18$ by $6$ .

$\ln (8 x + 9) = 3$

Step 3

To solve for $x$ , rewrite the equation using properties of logarithms.

$e^{\ln (8 x + 9)} = e^{3}$

Step 4

Rewrite $\ln (8 x + 9) = 3$ 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 $b^{y} = x$ .

$e^{3} = 8 x + 9$

Step 5

Solve for $x$ .

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Step 5.1

Rewrite the equation as $8 x + 9 = e^{3}$ .

$8 x + 9 = e^{3}$

Step 5.2

Subtract $9$ from both sides of the equation.

$8 x = e^{3} - 9$

Step 5.3

Divide each term in $8 x = e^{3} - 9$ by $8$ and simplify.

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Step 5.3.1

Divide each term in $8 x = e^{3} - 9$ by $8$ .

$\frac{8 x}{8} = \frac{e^{3}}{8} + \frac{- 9}{8}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $8$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{8 x}{8} = \frac{e^{3}}{8} + \frac{- 9}{8}$

Step 5.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{e^{3}}{8} + \frac{- 9}{8}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Move the negative in front of the fraction.

$x = \frac{e^{3}}{8} - \frac{9}{8}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{e^{3}}{8} - \frac{9}{8}$

Decimal Form:

$x = 1.38569211 \dots$