Trigonometry Examples

$\frac{1}{3} \cdot {(6)}^{- 4 x} + 1 = 6$

Step 1

Combine $\frac{1}{3}$ and $6^{- 4 x}$ .

$\frac{6^{- 4 x}}{3} + 1 = 6$

Step 2

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

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Step 2.1

Subtract $1$ from both sides of the equation.

$\frac{6^{- 4 x}}{3} = 6 - 1$

Step 2.2

Subtract $1$ from $6$ .

$\frac{6^{- 4 x}}{3} = 5$

Step 3

Multiply both sides by $3$ .

$\frac{6^{- 4 x}}{3} \cdot 3 = 5 \cdot 3$

Step 4

Simplify.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Cancel the common factor of $3$ .

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Step 4.1.1.1

Cancel the common factor.

$\frac{6^{- 4 x}}{3} \cdot 3 = 5 \cdot 3$

Step 4.1.1.2

Rewrite the expression.

$6^{- 4 x} = 5 \cdot 3$

Step 4.2

Simplify the right side.

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Step 4.2.1

Multiply $5$ by $3$ .

$6^{- 4 x} = 15$

Step 5

Solve for $x$ .

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Step 5.1

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

$\ln (6^{- 4 x}) = \ln (15)$

Step 5.2

Expand $\ln (6^{- 4 x})$ by moving $- 4 x$ outside the logarithm.

$- 4 x \ln (6) = \ln (15)$

Step 5.3

Divide each term in $- 4 x \ln (6) = \ln (15)$ by $- 4 \ln (6)$ and simplify.

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Step 5.3.1

Divide each term in $- 4 x \ln (6) = \ln (15)$ by $- 4 \ln (6)$ .

$\frac{- 4 x \ln (6)}{- 4 \ln (6)} = \frac{\ln (15)}{- 4 \ln (6)}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $- 4$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{- 4 x \ln (6)}{- 4 \ln (6)} = \frac{\ln (15)}{- 4 \ln (6)}$

Step 5.3.2.1.2

Rewrite the expression.

$\frac{x \ln (6)}{\ln (6)} = \frac{\ln (15)}{- 4 \ln (6)}$

Step 5.3.2.2

Cancel the common factor of $\ln (6)$ .

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Step 5.3.2.2.1

Cancel the common factor.

$\frac{x \ln (6)}{\ln (6)} = \frac{\ln (15)}{- 4 \ln (6)}$

Step 5.3.2.2.2

Divide $x$ by $1$ .

$x = \frac{\ln (15)}{- 4 \ln (6)}$

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.

$x = - \frac{\ln (15)}{4 \ln (6)}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{\ln (15)}{4 \ln (6)}$

Decimal Form:

$x = - 0.37784789 \dots$