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Answer :
To rewrite the expression [tex]\( 5^{-2} = \frac{1}{25} \)[/tex] in logarithmic form, we need to use the property of logarithms that relates an exponential equation to a logarithmic equation.
The property states that if [tex]\( a^b = c \)[/tex], then it can be rewritten in logarithmic form as [tex]\( \log_a(c) = b \)[/tex].
Here is a detailed step-by-step explanation using this property:
1. Identify the base, exponent, and result:
- The base [tex]\( a \)[/tex] in the exponentiation [tex]\( 5^{-2} = \frac{1}{25} \)[/tex] is [tex]\( 5 \)[/tex].
- The exponent [tex]\( b \)[/tex] is [tex]\( -2 \)[/tex].
- The result [tex]\( c \)[/tex] is [tex]\( \frac{1}{25} \)[/tex].
2. Apply the logarithmic property:
- Using the property [tex]\( a^b = c \)[/tex] translates to [tex]\( \log_a(c) = b \)[/tex].
- In this case, substitute [tex]\( 5 \)[/tex] for [tex]\( a \)[/tex], [tex]\( \frac{1}{25} \)[/tex] for [tex]\( c \)[/tex], and [tex]\( -2 \)[/tex] for [tex]\( b \)[/tex].
3. Write the logarithmic form:
- The equation [tex]\( 5^{-2} = \frac{1}{25} \)[/tex] can be rewritten as [tex]\( \log_5\left(\frac{1}{25}\right) = -2 \)[/tex].
This translates the exponential equation into its equivalent logarithmic form correctly.
The property states that if [tex]\( a^b = c \)[/tex], then it can be rewritten in logarithmic form as [tex]\( \log_a(c) = b \)[/tex].
Here is a detailed step-by-step explanation using this property:
1. Identify the base, exponent, and result:
- The base [tex]\( a \)[/tex] in the exponentiation [tex]\( 5^{-2} = \frac{1}{25} \)[/tex] is [tex]\( 5 \)[/tex].
- The exponent [tex]\( b \)[/tex] is [tex]\( -2 \)[/tex].
- The result [tex]\( c \)[/tex] is [tex]\( \frac{1}{25} \)[/tex].
2. Apply the logarithmic property:
- Using the property [tex]\( a^b = c \)[/tex] translates to [tex]\( \log_a(c) = b \)[/tex].
- In this case, substitute [tex]\( 5 \)[/tex] for [tex]\( a \)[/tex], [tex]\( \frac{1}{25} \)[/tex] for [tex]\( c \)[/tex], and [tex]\( -2 \)[/tex] for [tex]\( b \)[/tex].
3. Write the logarithmic form:
- The equation [tex]\( 5^{-2} = \frac{1}{25} \)[/tex] can be rewritten as [tex]\( \log_5\left(\frac{1}{25}\right) = -2 \)[/tex].
This translates the exponential equation into its equivalent logarithmic form correctly.
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