Trigonometry Examples

$4^{- 3 + x} = \frac{1}{18}$

Step 1

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

$\ln (4^{- 3 + x}) = \ln (\frac{1}{18})$

Step 2

Expand $\ln (4^{- 3 + x})$ by moving $- 3 + x$ outside the logarithm.

$(- 3 + x) \ln (4) = \ln (\frac{1}{18})$

Step 3

Simplify the left side.

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Step 3.1

Apply the distributive property.

$- 3 \ln (4) + x \ln (4) = \ln (\frac{1}{18})$

Step 4

Reorder $- 3 \ln (4)$ and $x \ln (4)$ .

$x \ln (4) - 3 \ln (4) = \ln (\frac{1}{18})$

Step 5

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

$x \ln (4) - 3 \ln (4) - \ln (\frac{1}{18}) = 0$

Step 6

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

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Step 6.1

Add $3 \ln (4)$ to both sides of the equation.

$x \ln (4) - \ln (\frac{1}{18}) = 3 \ln (4)$

Step 6.2

Add $\ln (\frac{1}{18})$ to both sides of the equation.

$x \ln (4) = 3 \ln (4) + \ln (\frac{1}{18})$

Step 7

Divide each term in $x \ln (4) = 3 \ln (4) + \ln (\frac{1}{18})$ by $\ln (4)$ and simplify.

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Step 7.1

Divide each term in $x \ln (4) = 3 \ln (4) + \ln (\frac{1}{18})$ by $\ln (4)$ .

$\frac{x \ln (4)}{\ln (4)} = \frac{3 \ln (4)}{\ln (4)} + \frac{\ln (\frac{1}{18})}{\ln (4)}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $\ln (4)$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{x \ln (4)}{\ln (4)} = \frac{3 \ln (4)}{\ln (4)} + \frac{\ln (\frac{1}{18})}{\ln (4)}$

Step 7.2.1.2

Divide $x$ by $1$ .

$x = \frac{3 \ln (4)}{\ln (4)} + \frac{\ln (\frac{1}{18})}{\ln (4)}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Cancel the common factor of $\ln (4)$ .

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Step 7.3.1.1

Cancel the common factor.

$x = \frac{3 \ln (4)}{\ln (4)} + \frac{\ln (\frac{1}{18})}{\ln (4)}$

Step 7.3.1.2

Divide $3$ by $1$ .

$x = 3 + \frac{\ln (\frac{1}{18})}{\ln (4)}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = 3 + \frac{\ln (\frac{1}{18})}{\ln (4)}$

Decimal Form:

$x = 0.91503749 \dots$