Trigonometry Examples

$10^{7 - x} = 470$

Step 1

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

$\log (10^{7 - x}) = \log (470)$

Step 2

Expand the left side.

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Step 2.1

Expand $\log (10^{7 - x})$ by moving $7 - x$ outside the logarithm.

$(7 - x) \log (10) = \log (470)$

Step 2.2

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

$(7 - x) \cdot 1 = \log (470)$

Step 2.3

Multiply $7 - x$ by $1$ .

$7 - x = \log (470)$

Step 3

Subtract $7$ from both sides of the equation.

$- x = \log (470) - 7$

Step 4

Divide each term in $- x = \log (470) - 7$ by $- 1$ and simplify.

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Step 4.1

Divide each term in $- x = \log (470) - 7$ by $- 1$ .

$\frac{- x}{- 1} = \frac{\log (470)}{- 1} + \frac{- 7}{- 1}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{\log (470)}{- 1} + \frac{- 7}{- 1}$

Step 4.2.2

Divide $x$ by $1$ .

$x = \frac{\log (470)}{- 1} + \frac{- 7}{- 1}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Move the negative one from the denominator of $\frac{\log (470)}{- 1}$ .

$x = - 1 \cdot \log (470) + \frac{- 7}{- 1}$

Step 4.3.1.2

Rewrite $- 1 \cdot \log (470)$ as $- \log (470)$ .

$x = - \log (470) + \frac{- 7}{- 1}$

Step 4.3.1.3

Divide $- 7$ by $- 1$ .

$x = - \log (470) + 7$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - \log (470) + 7$

Decimal Form:

$x = 4.32790214 \dots$