Trigonometry Examples

$- 2 \cdot 13^{3 x - 9} + 8 = - 69$

Step 1

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

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Step 1.1

Subtract $8$ from both sides of the equation.

$- 2 \cdot 13^{3 x - 9} = - 69 - 8$

Step 1.2

Subtract $8$ from $- 69$ .

$- 2 \cdot 13^{3 x - 9} = - 77$

Step 2

Divide each term in $- 2 \cdot 13^{3 x - 9} = - 77$ by $- 2$ and simplify.

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Step 2.1

Divide each term in $- 2 \cdot 13^{3 x - 9} = - 77$ by $- 2$ .

$\frac{- 2 \cdot 13^{3 x - 9}}{- 2} = \frac{- 77}{- 2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $- 2$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{- 2 \cdot 13^{3 x - 9}}{- 2} = \frac{- 77}{- 2}$

Step 2.2.1.2

Divide $13^{3 x - 9}$ by $1$ .

$13^{3 x - 9} = \frac{- 77}{- 2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Dividing two negative values results in a positive value.

$13^{3 x - 9} = \frac{77}{2}$

Step 3

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (13^{3 x - 9}) = \ln (\frac{77}{2})$

Step 4

Expand $\ln (13^{3 x - 9})$ by moving $3 x - 9$ outside the logarithm.

$(3 x - 9) \ln (13) = \ln (\frac{77}{2})$

Step 5

Simplify the left side.

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Step 5.1

Apply the distributive property.

$3 x \ln (13) - 9 \ln (13) = \ln (\frac{77}{2})$

Step 6

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

$3 x \ln (13) - 9 \ln (13) - \ln (\frac{77}{2}) = 0$

Step 7

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

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Step 7.1

Add $9 \ln (13)$ to both sides of the equation.

$3 x \ln (13) - \ln (\frac{77}{2}) = 9 \ln (13)$

Step 7.2

Add $\ln (\frac{77}{2})$ to both sides of the equation.

$3 x \ln (13) = 9 \ln (13) + \ln (\frac{77}{2})$

Step 8

Divide each term in $3 x \ln (13) = 9 \ln (13) + \ln (\frac{77}{2})$ by $3 \ln (13)$ and simplify.

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Step 8.1

Divide each term in $3 x \ln (13) = 9 \ln (13) + \ln (\frac{77}{2})$ by $3 \ln (13)$ .

$\frac{3 x \ln (13)}{3 \ln (13)} = \frac{9 \ln (13)}{3 \ln (13)} + \frac{\ln (\frac{77}{2})}{3 \ln (13)}$

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $3$ .

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Step 8.2.1.1

Cancel the common factor.

$\frac{3 x \ln (13)}{3 \ln (13)} = \frac{9 \ln (13)}{3 \ln (13)} + \frac{\ln (\frac{77}{2})}{3 \ln (13)}$

Step 8.2.1.2

Rewrite the expression.

$\frac{x \ln (13)}{\ln (13)} = \frac{9 \ln (13)}{3 \ln (13)} + \frac{\ln (\frac{77}{2})}{3 \ln (13)}$

Step 8.2.2

Cancel the common factor of $\ln (13)$ .

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Step 8.2.2.1

Cancel the common factor.

$\frac{x \ln (13)}{\ln (13)} = \frac{9 \ln (13)}{3 \ln (13)} + \frac{\ln (\frac{77}{2})}{3 \ln (13)}$

Step 8.2.2.2

Divide $x$ by $1$ .

$x = \frac{9 \ln (13)}{3 \ln (13)} + \frac{\ln (\frac{77}{2})}{3 \ln (13)}$

Step 8.3

Simplify the right side.

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Step 8.3.1

Cancel the common factor of $9$ and $3$ .

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Step 8.3.1.1

Factor $3$ out of $9 \ln (13)$ .

$x = \frac{3 (3 \ln (13))}{3 \ln (13)} + \frac{\ln (\frac{77}{2})}{3 \ln (13)}$

Step 8.3.1.2

Cancel the common factor.

$x = \frac{3 (3 \ln (13))}{3 \ln (13)} + \frac{\ln (\frac{77}{2})}{3 \ln (13)}$

Step 8.3.1.3

Rewrite the expression.

$x = 3 + \frac{\ln (\frac{77}{2})}{3 \ln (13)}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = 3 + \frac{\ln (\frac{77}{2})}{3 \ln (13)}$

Decimal Form:

$x = 3.47442889 \dots$