To determine which logarithmic equation is equivalent to the exponential equation [tex]\( 5^c = 250 \)[/tex], let's recall how logarithms work.
The exponential equation [tex]\( a^b = c \)[/tex] can be converted to its equivalent logarithmic form using the relationship:
[tex]\[ b = \log_a c. \][/tex]
Given the exponential equation:
[tex]\[ 5^c = 250, \][/tex]
we need to express this in the logarithmic form. According to the rule for logarithms, the exponent [tex]\( c \)[/tex] must be isolated, and the base [tex]\( 5 \)[/tex] and the result [tex]\( 250 \)[/tex] are set up in a logarithmic expression.
Following the conversion rule:
[tex]\[ c = \log_5 250. \][/tex]
Upon examining the given options, we find:
A. [tex]\(\log_5 250 = c\)[/tex]
B. [tex]\(\log_5 c = 250\)[/tex]
C. [tex]\(\log_{250} c = 5\)[/tex]
D. [tex]\(\log_c 250 = 5\)[/tex]
From our logarithmic conversion, we see that option A ([tex]\(\log_5 250 = c\)[/tex]) accurately represents the relationship derived from the exponential equation [tex]\( 5^c = 250 \)[/tex].
Thus, the correct answer is:
A. [tex]\(\log_5 250 = c\)[/tex].
