Precalculus Examples

$\log (\frac{1}{10^{- 3}}) + \log (10^{4})$

Step 1

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\log (\frac{1}{10^{- 3}} \cdot 10^{4})$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\log (\frac{1}{\frac{1}{10^{3}}} \cdot 10^{4})$

Step 2.2

Raise $10$ to the power of $3$ .

$\log (\frac{1}{\frac{1}{1000}} \cdot 10^{4})$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\log (1 \cdot 1000 \cdot 10^{4})$

Step 4

Multiply $1000$ by $1$ .

$\log (1000 \cdot 10^{4})$

Step 5

Move the decimal point in $1000$ to the left by $3$ places and increase the power of $10^{4}$ by $3$ .

$\log (1 \cdot 10^{7})$

Step 6

Multiply $10^{7}$ by $1$ .

$\log (10^{7})$

Step 7

Use logarithm rules to move $7$ out of the exponent.

$7 \log (10)$

Step 8

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

$7 \cdot 1$

Step 9

Multiply $7$ by $1$ .

$7$