Trigonometry Examples

$- 4 \cdot 10^{6 k - 10} + 5 = - 73$

Step 1

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

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Step 1.1

Subtract $5$ from both sides of the equation.

$- 4 \cdot 10^{6 k - 10} = - 73 - 5$

Step 1.2

Subtract $5$ from $- 73$ .

$- 4 \cdot 10^{6 k - 10} = - 78$

Step 2

Divide each term in $- 4 \cdot 10^{6 k - 10} = - 78$ by $- 4$ and simplify.

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Step 2.1

Divide each term in $- 4 \cdot 10^{6 k - 10} = - 78$ by $- 4$ .

$\frac{- 4 \cdot 10^{6 k - 10}}{- 4} = \frac{- 78}{- 4}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $- 4$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{- 4 \cdot 10^{6 k - 10}}{- 4} = \frac{- 78}{- 4}$

Step 2.2.1.2

Divide $10^{6 k - 10}$ by $1$ .

$10^{6 k - 10} = \frac{- 78}{- 4}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Cancel the common factor of $- 78$ and $- 4$ .

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Step 2.3.1.1

Factor $- 2$ out of $- 78$ .

$10^{6 k - 10} = \frac{- 2 (39)}{- 4}$

Step 2.3.1.2

Cancel the common factors.

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Step 2.3.1.2.1

Factor $- 2$ out of $- 4$ .

$10^{6 k - 10} = \frac{- 2 \cdot 39}{- 2 \cdot 2}$

Step 2.3.1.2.2

Cancel the common factor.

$10^{6 k - 10} = \frac{- 2 \cdot 39}{- 2 \cdot 2}$

Step 2.3.1.2.3

Rewrite the expression.

$10^{6 k - 10} = \frac{39}{2}$

Step 3

Take the base $10$ logarithm of both sides of the equation to remove the variable from the exponent.

$\log (10^{6 k - 10}) = \log (\frac{39}{2})$

Step 4

Expand the left side.

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Step 4.1

Expand $\log (10^{6 k - 10})$ by moving $6 k - 10$ outside the logarithm.

$(6 k - 10) \log (10) = \log (\frac{39}{2})$

Step 4.2

Logarithm base $10$ of $10$ is $1$ .

$(6 k - 10) \cdot 1 = \log (\frac{39}{2})$

Step 4.3

Multiply $6 k - 10$ by $1$ .

$6 k - 10 = \log (\frac{39}{2})$

Step 5

Add $10$ to both sides of the equation.

$6 k = \log (\frac{39}{2}) + 10$

Step 6

Divide each term in $6 k = \log (\frac{39}{2}) + 10$ by $6$ and simplify.

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Step 6.1

Divide each term in $6 k = \log (\frac{39}{2}) + 10$ by $6$ .

$\frac{6 k}{6} = \frac{\log (\frac{39}{2})}{6} + \frac{10}{6}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $6$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{6 k}{6} = \frac{\log (\frac{39}{2})}{6} + \frac{10}{6}$

Step 6.2.1.2

Divide $k$ by $1$ .

$k = \frac{\log (\frac{39}{2})}{6} + \frac{10}{6}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Cancel the common factor of $10$ and $6$ .

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Step 6.3.1.1

Factor $2$ out of $10$ .

$k = \frac{\log (\frac{39}{2})}{6} + \frac{2 (5)}{6}$

Step 6.3.1.2

Cancel the common factors.

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Step 6.3.1.2.1

Factor $2$ out of $6$ .

$k = \frac{\log (\frac{39}{2})}{6} + \frac{2 \cdot 5}{2 \cdot 3}$

Step 6.3.1.2.2

Cancel the common factor.

$k = \frac{\log (\frac{39}{2})}{6} + \frac{2 \cdot 5}{2 \cdot 3}$

Step 6.3.1.2.3

Rewrite the expression.

$k = \frac{\log (\frac{39}{2})}{6} + \frac{5}{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$k = \frac{\log (\frac{39}{2})}{6} + \frac{5}{3}$

Decimal Form:

$k = 1.88167243 \dots$