College

Express the following equations in logarithmic form:

(a) [tex]\(5^5 = 3125\)[/tex] is equivalent to the logarithmic equation: [tex]\(\log_5 3125 = \square\)[/tex]

(b) [tex]\(10^{-2} = 0.01\)[/tex] is equivalent to the logarithmic equation: [tex]\(\log_{10} 0.01 = \square\)[/tex]

Answer :

Sure, let's express the given exponential equations in their equivalent logarithmic form.

When we have an equation in the form [tex]\( a^b = c \)[/tex], we can write it in logarithmic form as [tex]\( \log_a(c) = b \)[/tex]. Here, the base [tex]\( a \)[/tex] of the exponential becomes the base of the logarithm, the exponent [tex]\( b \)[/tex] becomes the result of the logarithm, and the result [tex]\( c \)[/tex] of the exponential becomes the number we are taking the logarithm of.

Let's apply this to the two given equations:

(a) [tex]\( 5^5 = 3125 \)[/tex]

Using the logarithmic form:
- Base of the exponential is 5.
- Exponent is 5.
- Result is 3125.

So the logarithmic form of this equation is [tex]\(\log_5(3125) = 5\)[/tex].

(b) [tex]\( 10^{-2} = 0.01 \)[/tex]

Using the logarithmic form:
- Base of the exponential is 10.
- Exponent is -2.
- Result is 0.01.

So the logarithmic form of this equation is [tex]\(\log_{10}(0.01) = -2\)[/tex].

Therefore, the logarithmic forms are:

(a) [tex]\(\log_5(3125) = 5\)[/tex]

(b) [tex]\(\log_{10}(0.01) = -2\)[/tex]

These are the equivalent logarithmic equations for the given exponential equations.