Answer :
The value of [tex]\( \log_5 \frac{1}{25}[/tex] is: -2
To evaluate [tex]\( \log_5 \frac{1}{25} \)[/tex] without a calculator, we can use the properties of logarithms and exponents to simplify the expression. Here's the step-by-step solution:
1. Recall the definition of a logarithm: if [tex]\( a^x = b \)[/tex], then [tex]\( \log_a b = x \)[/tex].
2. We need to express [tex]\( \frac{1}{25} \)[/tex] as a power of 5 because the base of our logarithm is 5.
Notice that [tex]\( 5^2 = 25 \)[/tex]. Therefore, [tex]\( 5^{-2} \)[/tex] is [tex]\( \frac{1}{5^2} \)[/tex], which equals [tex]\( \frac{1}{25} \)[/tex].
Now we can rewrite the original logarithmic expression using the property we've identified:
[tex]\[ \log_5 \frac{1}{25} = \log_5 5^{-2} \][/tex]
3. The next step is to use the property of logarithms that says [tex]\( \log_a a^x = x \)[/tex]. Since we have [tex]\( \log_5 5^{-2} \)[/tex], we can simply extract the exponent of 5, which is -2.
[tex]\[ \log_5 5^{-2} = -2 \][/tex]
[tex]\( \log_5 \frac{1}{25} = -2 \).[/tex]