Trigonometry Examples

$4 \cdot 2^{n - 46} = 97$

Step 1

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

$\ln (4 \cdot 2^{n - 46}) = \ln (97)$

Step 2

Expand the left side.

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Step 2.1

Rewrite $\ln (4 \cdot 2^{n - 46})$ as $\ln (4) + \ln (2^{n - 46})$ .

$\ln (4) + \ln (2^{n - 46}) = \ln (97)$

Step 2.2

Expand $\ln (2^{n - 46})$ by moving $n - 46$ outside the logarithm.

$\ln (4) + (n - 46) \ln (2) = \ln (97)$

Step 3

Simplify the left side.

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Step 3.1

Apply the distributive property.

$\ln (4) + n \ln (2) - 46 \ln (2) = \ln (97)$

Step 4

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

$n \ln (2) + \ln (4) - 46 \ln (2) = \ln (97)$

Step 5

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

$n \ln (2) + \ln (4) - 46 \ln (2) - \ln (97) = 0$

Step 6

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

$n \ln (2) + \ln (\frac{4}{97}) - 46 \ln (2) = 0$

Step 7

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

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Step 7.1

Subtract $\ln (\frac{4}{97})$ from both sides of the equation.

$n \ln (2) - 46 \ln (2) = - \ln (\frac{4}{97})$

Step 7.2

Add $46 \ln (2)$ to both sides of the equation.

$n \ln (2) = - \ln (\frac{4}{97}) + 46 \ln (2)$

Step 8

Divide each term in $n \ln (2) = - \ln (\frac{4}{97}) + 46 \ln (2)$ by $\ln (2)$ and simplify.

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Step 8.1

Divide each term in $n \ln (2) = - \ln (\frac{4}{97}) + 46 \ln (2)$ by $\ln (2)$ .

$\frac{n \ln (2)}{\ln (2)} = \frac{- \ln (\frac{4}{97})}{\ln (2)} + \frac{46 \ln (2)}{\ln (2)}$

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.

$\frac{n \ln (2)}{\ln (2)} = \frac{- \ln (\frac{4}{97})}{\ln (2)} + \frac{46 \ln (2)}{\ln (2)}$

Step 8.2.1.2

Divide $n$ by $1$ .

$n = \frac{- \ln (\frac{4}{97})}{\ln (2)} + \frac{46 \ln (2)}{\ln (2)}$

Step 8.3

Simplify the right side.

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Step 8.3.1

Simplify each term.

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Step 8.3.1.1

Move the negative in front of the fraction.

$n = - \frac{\ln (\frac{4}{97})}{\ln (2)} + \frac{46 \ln (2)}{\ln (2)}$

Step 8.3.1.2

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

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Step 8.3.1.2.1

Cancel the common factor.

$n = - \frac{\ln (\frac{4}{97})}{\ln (2)} + \frac{46 \ln (2)}{\ln (2)}$

Step 8.3.1.2.2

Divide $46$ by $1$ .

$n = - \frac{\ln (\frac{4}{97})}{\ln (2)} + 46$

Step 9

The result can be shown in multiple forms.

Exact Form:

$n = - \frac{\ln (\frac{4}{97})}{\ln (2)} + 46$

Decimal Form:

$n = 50.59991284 \dots$