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Answer :
To write the equation [tex]\(\frac{1}{25} = 5^{-2}\)[/tex] in logarithmic form, we need to follow these steps:
1. Understand the Exponential Form:
In the given equation, [tex]\(\frac{1}{25} = 5^{-2}\)[/tex], the base is [tex]\(5\)[/tex], the exponent is [tex]\(-2\)[/tex], and the result is [tex]\(\frac{1}{25}\)[/tex].
2. Recall the Logarithmic Form:
The general form of converting an exponential equation [tex]\(a^b = c\)[/tex] into logarithmic form is [tex]\(\log_a(c) = b\)[/tex]. This means we are finding the exponent [tex]\(b\)[/tex] to which the base [tex]\(a\)[/tex] must be raised to produce the number [tex]\(c\)[/tex].
3. Identify Components:
- Base [tex]\(a\)[/tex] is [tex]\(5\)[/tex].
- The result [tex]\(c\)[/tex] is [tex]\(\frac{1}{25}\)[/tex].
- The exponent [tex]\(b\)[/tex] is [tex]\(-2\)[/tex].
4. Write the Equation in Logarithmic Form:
Substitute the identified components into the logarithmic formula:
[tex]\[
\log_5\left(\frac{1}{25}\right) = -2
\][/tex]
Therefore, the logarithmic form of the equation [tex]\(\frac{1}{25} = 5^{-2}\)[/tex] is [tex]\(\log_5\left(\frac{1}{25}\right) = -2\)[/tex].
1. Understand the Exponential Form:
In the given equation, [tex]\(\frac{1}{25} = 5^{-2}\)[/tex], the base is [tex]\(5\)[/tex], the exponent is [tex]\(-2\)[/tex], and the result is [tex]\(\frac{1}{25}\)[/tex].
2. Recall the Logarithmic Form:
The general form of converting an exponential equation [tex]\(a^b = c\)[/tex] into logarithmic form is [tex]\(\log_a(c) = b\)[/tex]. This means we are finding the exponent [tex]\(b\)[/tex] to which the base [tex]\(a\)[/tex] must be raised to produce the number [tex]\(c\)[/tex].
3. Identify Components:
- Base [tex]\(a\)[/tex] is [tex]\(5\)[/tex].
- The result [tex]\(c\)[/tex] is [tex]\(\frac{1}{25}\)[/tex].
- The exponent [tex]\(b\)[/tex] is [tex]\(-2\)[/tex].
4. Write the Equation in Logarithmic Form:
Substitute the identified components into the logarithmic formula:
[tex]\[
\log_5\left(\frac{1}{25}\right) = -2
\][/tex]
Therefore, the logarithmic form of the equation [tex]\(\frac{1}{25} = 5^{-2}\)[/tex] is [tex]\(\log_5\left(\frac{1}{25}\right) = -2\)[/tex].
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