Precalculus Examples

$2 \cdot 10^{2 - x} = 13$

Step 1

Divide each term in $2 \cdot 10^{2 - x} = 13$ by $2$ and simplify.

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Step 1.1

Divide each term in $2 \cdot 10^{2 - x} = 13$ by $2$ .

$\frac{2 \cdot 10^{2 - x}}{2} = \frac{13}{2}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $2$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{2 \cdot 10^{2 - x}}{2} = \frac{13}{2}$

Step 1.2.1.2

Divide $10^{2 - x}$ by $1$ .

$10^{2 - x} = \frac{13}{2}$

Step 2

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

$\log (10^{2 - x}) = \log (\frac{13}{2})$

Step 3

Expand the left side.

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Step 3.1

Expand $\log (10^{2 - x})$ by moving $2 - x$ outside the logarithm.

$(2 - x) \log (10) = \log (\frac{13}{2})$

Step 3.2

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

$(2 - x) \cdot 1 = \log (\frac{13}{2})$

Step 3.3

Multiply $2 - x$ by $1$ .

$2 - x = \log (\frac{13}{2})$

Step 4

Subtract $2$ from both sides of the equation.

$- x = \log (\frac{13}{2}) - 2$

Step 5

Divide each term in $- x = \log (\frac{13}{2}) - 2$ by $- 1$ and simplify.

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Step 5.1

Divide each term in $- x = \log (\frac{13}{2}) - 2$ by $- 1$ .

$\frac{- x}{- 1} = \frac{\log (\frac{13}{2})}{- 1} + \frac{- 2}{- 1}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{\log (\frac{13}{2})}{- 1} + \frac{- 2}{- 1}$

Step 5.2.2

Divide $x$ by $1$ .

$x = \frac{\log (\frac{13}{2})}{- 1} + \frac{- 2}{- 1}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Simplify each term.

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Step 5.3.1.1

Move the negative one from the denominator of $\frac{\log (\frac{13}{2})}{- 1}$ .

$x = - 1 \cdot \log (\frac{13}{2}) + \frac{- 2}{- 1}$

Step 5.3.1.2

Rewrite $- 1 \cdot \log (\frac{13}{2})$ as $- \log (\frac{13}{2})$ .

$x = - \log (\frac{13}{2}) + \frac{- 2}{- 1}$

Step 5.3.1.3

Divide $- 2$ by $- 1$ .

$x = - \log (\frac{13}{2}) + 2$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = - \log (\frac{13}{2}) + 2$

Decimal Form:

$x = 1.18708664 \dots$