Precalculus Examples

$10 \log (\frac{2 \cdot 10^{x}}{1 \cdot 10^{- 12}})$

Step 1

Cancel the common factor of $10^{x}$ and $10^{- 12}$ .

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Step 1.1

Factor $10^{- 12}$ out of $2 \cdot 10^{x}$ .

$10 \log (\frac{10^{- 12} (2 \cdot 10^{x + 12})}{1 \cdot 10^{- 12}})$

Step 1.2

Cancel the common factors.

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Step 1.2.1

Factor $10^{- 12}$ out of $1 \cdot 10^{- 12}$ .

$10 \log (\frac{10^{- 12} (2 \cdot 10^{x + 12})}{10^{- 12} \cdot 1})$

Step 1.2.2

Cancel the common factor.

$10 \log (\frac{10^{- 12} (2 \cdot 10^{x + 12})}{10^{- 12} \cdot 1})$

Step 1.2.3

Rewrite the expression.

$10 \log (\frac{2 \cdot 10^{x + 12}}{1})$

Step 1.2.4

Divide $2 \cdot 10^{x + 12}$ by $1$ .

$10 \log (2 \cdot 10^{x + 12})$

Step 2

Rewrite $\log (2 \cdot 10^{x + 12})$ as $\log (2) + \log (10^{x + 12})$ .

$10 (\log (2) + \log (10^{x + 12}))$

Step 3

Use logarithm rules to move $x + 12$ out of the exponent.

$10 (\log (2) + (x + 12) \log (10))$

Step 4

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

$10 (\log (2) + (x + 12) \cdot 1)$

Step 5

Multiply $x + 12$ by $1$ .

$10 (\log (2) + x + 12)$

Step 6

Apply the distributive property.

$10 \log (2) + 10 x + 10 \cdot 12$

Step 7

Simplify.

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Step 7.1

Simplify $10 \log (2)$ by moving $10$ inside the logarithm.

$\log (2^{10}) + 10 x + 10 \cdot 12$

Step 7.2

Multiply $10$ by $12$ .

$\log (2^{10}) + 10 x + 120$

Step 8

Raise $2$ to the power of $10$ .

$\log (1024) + 10 x + 120$