Convert to the logarithmic equation that has the same meaning.
[tex]$3125^{\frac{1}{5}}=5$[/tex]
Answer (1)
Recognize the exponential form b x = y .
Identify the base, exponent, and result: b = 3125 , x = 5 1 , y = 5 .
Convert to logarithmic form using lo g b y = x .
The logarithmic equation is lo g 3125 5 = 5 1 .
Explanation
Understanding the Problem We are given the exponential equation 312 5 5 1 = 5 and asked to convert it into its equivalent logarithmic form.
Recalling the Conversion Formula Recall that an exponential equation of the form b x = y can be rewritten in logarithmic form as lo g b y = x , where b is the base, x is the exponent, and y is the result.
Identifying Base, Exponent, and Result In our equation, 312 5 5 1 = 5 , we can identify the base as b = 3125 , the exponent as x = 5 1 , and the result as y = 5 .
Converting to Logarithmic Form Substituting these values into the logarithmic form lo g b y = x , we get lo g 3125 5 = 5 1 . This is the logarithmic equation that has the same meaning as the given exponential equation.
Examples
Logarithmic equations are used in various fields, such as calculating the magnitude of earthquakes on the Richter scale, determining the pH of a solution in chemistry, and modeling population growth in biology. For instance, if we know that the intensity of an earthquake is 100,000 times the reference intensity, we can use logarithms to find its magnitude on the Richter scale: M = lo g 10 ( 100 , 000 ) = 5 .
