To find the exact value of [tex]\(\log 10^{-5}\)[/tex], we can utilize the properties of logarithms and exponents.
### Step-by-Step Solution:
1. Consider the expression [tex]\(\log 10^{-5}\)[/tex]:
[tex]\[\log 10^{-5}\][/tex]
2. Use the property of logarithms dealing with exponents:
The logarithm of a power can be rewritten by bringing the exponent in front of the log.
[tex]\[\log 10^{-5} = -5 \log 10\][/tex]
3. Evaluate [tex]\(\log 10\)[/tex] with base 10:
By definition, [tex]\(\log 10\)[/tex] in base 10 is equal to 1.
[tex]\[\log 10 = 1\][/tex]
4. Multiply the exponent by the value of [tex]\(\log 10\)[/tex]:
Substitute the value of [tex]\(\log 10\)[/tex] back into the expression.
[tex]\[-5 \log 10 = -5 \times 1\][/tex]
5. Calculate the result:
[tex]\[-5 \times 1 = -5\][/tex]
Therefore, the exact value of [tex]\(\log 10^{-5}\)[/tex] is:
[tex]\[
\boxed{-5}
\][/tex]
