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Answer :
To express exponential equations in logarithmic form, we use the basic relationship between exponents and logarithms. If [tex]\( a^b = c \)[/tex], then the equivalent logarithmic form is [tex]\( \log_a(c) = b \)[/tex]. Let's apply this concept to the given equations:
(a) Convert [tex]\( 5^5 = 3125 \)[/tex] to logarithmic form:
1. Identify the base of the exponent. Here, the base is 5.
2. The exponent is 5, and the result of the exponentiation is 3125.
3. According to the relationship, [tex]\( a^b = c \)[/tex] converts to [tex]\( \log_a(c) = b \)[/tex].
4. Therefore, the logarithmic form is [tex]\( \log_5(3125) = 5 \)[/tex].
(b) Convert [tex]\( 10^{-4} = 0.0001 \)[/tex] to logarithmic form:
1. Identify the base of the exponent. Here, the base is 10.
2. The exponent is [tex]\(-4\)[/tex], and the result is 0.0001.
3. Using the same relationship as before, [tex]\( a^b = c \)[/tex] becomes [tex]\( \log_a(c) = b \)[/tex].
4. Thus, the logarithmic form is [tex]\( \log_{10}(0.0001) = -4 \)[/tex].
These transformations show how to take an equation from exponential form into its equivalent logarithmic form.
(a) Convert [tex]\( 5^5 = 3125 \)[/tex] to logarithmic form:
1. Identify the base of the exponent. Here, the base is 5.
2. The exponent is 5, and the result of the exponentiation is 3125.
3. According to the relationship, [tex]\( a^b = c \)[/tex] converts to [tex]\( \log_a(c) = b \)[/tex].
4. Therefore, the logarithmic form is [tex]\( \log_5(3125) = 5 \)[/tex].
(b) Convert [tex]\( 10^{-4} = 0.0001 \)[/tex] to logarithmic form:
1. Identify the base of the exponent. Here, the base is 10.
2. The exponent is [tex]\(-4\)[/tex], and the result is 0.0001.
3. Using the same relationship as before, [tex]\( a^b = c \)[/tex] becomes [tex]\( \log_a(c) = b \)[/tex].
4. Thus, the logarithmic form is [tex]\( \log_{10}(0.0001) = -4 \)[/tex].
These transformations show how to take an equation from exponential form into its equivalent logarithmic form.
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