Trigonometry Examples

$4 (2^{x}) = 6$

Step 1

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

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

Step 2

Expand the left side.

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Step 2.1

Rewrite $\ln (4 \cdot 2^{x})$ as $\ln (4) + \ln (2^{x})$ .

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

Step 2.2

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

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

Step 3

Reorder $\ln (4)$ and $x \ln (2)$ .

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

Step 4

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

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

Step 5

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$x \ln (2) + \ln (\frac{4}{6}) = 0$

Step 6

Cancel the common factor of $4$ and $6$ .

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Step 6.1

Factor $2$ out of $4$ .

$x \ln (2) + \ln (\frac{2 (2)}{6}) = 0$

Step 6.2

Cancel the common factors.

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Step 6.2.1

Factor $2$ out of $6$ .

$x \ln (2) + \ln (\frac{2 \cdot 2}{2 \cdot 3}) = 0$

Step 6.2.2

Cancel the common factor.

$x \ln (2) + \ln (\frac{2 \cdot 2}{2 \cdot 3}) = 0$

Step 6.2.3

Rewrite the expression.

$x \ln (2) + \ln (\frac{2}{3}) = 0$

Step 7

Subtract $\ln (\frac{2}{3})$ from both sides of the equation.

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

Step 8

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

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Step 8.1

Divide each term in $x \ln (2) = - \ln (\frac{2}{3})$ by $\ln (2)$ .

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

Step 8.2

Simplify the left side.

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Step 8.2.1

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

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Step 8.2.1.1

Cancel the common factor.

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

Step 8.2.1.2

Divide $x$ by $1$ .

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

Step 8.3

Simplify the right side.

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Step 8.3.1

Move the negative in front of the fraction.

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

Step 9

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 0.58496250 \dots$