Question

Using numerical analysis and the Power Rule of Logarithms,
log
x
k
=
k
log
x
, how can
log
(
1
100
)
be rewritten?(1 point)
Responses

−
2
log
10
negative 2 logarithm 10

log
−
20
logarithm negative 20

10
log
−
2
10 logarithm negative 2

2
log
−
10

Answer 1

To rewrite \( \log\left(\frac{1}{100}\right) \) using the Power Rule of Logarithms, we first recognize that \( \frac{1}{100} \) can be expressed as \( 100^{-1} \).

Applying the Power Rule, which states that \( \log\left(x^k\right) = k \cdot \log(x) \), we have:

\[
\log\left(\frac{1}{100}\right) = \log\left(100^{-1}\right) = -1 \cdot \log(100)
\]

Now we can further simplify \( \log(100) \). Since \( 100 = 10^2 \), we can apply the Power Rule again:

\[
\log(100) = \log(10^2) = 2 \cdot \log(10)
\]

Now substituting this back into our earlier expression gives:

\[
\log\left(\frac{1}{100}\right) = -1 \cdot (2 \cdot \log(10)) = -2 \cdot \log(10)
\]

The result is:

\[
\log\left(\frac{1}{100}\right) = -2 \log(10)
\]

Thus, the correct response is:

**−2 log 10**.