Trigonometry Examples

$3 {(2)}^{x - 4} = 18$

Step 1

Divide each term in $3 \cdot 2^{x - 4} = 18$ by $3$ and simplify.

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Step 1.1

Divide each term in $3 \cdot 2^{x - 4} = 18$ by $3$ .

$\frac{3 \cdot 2^{x - 4}}{3} = \frac{18}{3}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $3$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{3 \cdot 2^{x - 4}}{3} = \frac{18}{3}$

Step 1.2.1.2

Divide $2^{x - 4}$ by $1$ .

$2^{x - 4} = \frac{18}{3}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Divide $18$ by $3$ .

$2^{x - 4} = 6$

Step 2

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

$\ln (2^{x - 4}) = \ln (6)$

Step 3

Expand $\ln (2^{x - 4})$ by moving $x - 4$ outside the logarithm.

$(x - 4) \ln (2) = \ln (6)$

Step 4

Simplify the left side.

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Step 4.1

Apply the distributive property.

$x \ln (2) - 4 \ln (2) = \ln (6)$

Step 5

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

$x \ln (2) - 4 \ln (2) - \ln (6) = 0$

Step 6

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

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Step 6.1

Add $4 \ln (2)$ to both sides of the equation.

$x \ln (2) - \ln (6) = 4 \ln (2)$

Step 6.2

Add $\ln (6)$ to both sides of the equation.

$x \ln (2) = 4 \ln (2) + \ln (6)$

Step 7

Divide each term in $x \ln (2) = 4 \ln (2) + \ln (6)$ by $\ln (2)$ and simplify.

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Step 7.1

Divide each term in $x \ln (2) = 4 \ln (2) + \ln (6)$ by $\ln (2)$ .

$\frac{x \ln (2)}{\ln (2)} = \frac{4 \ln (2)}{\ln (2)} + \frac{\ln (6)}{\ln (2)}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $\ln (2)$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{x \ln (2)}{\ln (2)} = \frac{4 \ln (2)}{\ln (2)} + \frac{\ln (6)}{\ln (2)}$

Step 7.2.1.2

Divide $x$ by $1$ .

$x = \frac{4 \ln (2)}{\ln (2)} + \frac{\ln (6)}{\ln (2)}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Cancel the common factor of $\ln (2)$ .

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Step 7.3.1.1

Cancel the common factor.

$x = \frac{4 \ln (2)}{\ln (2)} + \frac{\ln (6)}{\ln (2)}$

Step 7.3.1.2

Divide $4$ by $1$ .

$x = 4 + \frac{\ln (6)}{\ln (2)}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = 4 + \frac{\ln (6)}{\ln (2)}$

Decimal Form:

$x = 6.58496250 \dots$