$\log _{\frac{1}{5}} 5=$
Answer (1)
We are given the logarithmic expression lo g 5 1 5 .
Convert the logarithm to exponential form: ( 5 1 ) x = 5 .
Rewrite the equation with the same base: 5 − x = 5 1 .
Solve for x : x = − 1 , so the final answer is − 1 .
Explanation
Understanding the Problem We are asked to evaluate the logarithmic expression lo g 5 1 5 . This means we need to find the exponent to which we must raise 5 1 to get 5.
Converting to Exponential Form Let x = lo g 5 1 5 . We can rewrite this logarithmic equation in exponential form as ( 5 1 ) x = 5 .
Expressing with the Same Base We can express 5 1 as 5 − 1 , so our equation becomes ( 5 − 1 ) x = 5 . This simplifies to 5 − x = 5 1 .
Equating Exponents Since the bases are equal, we can equate the exponents: − x = 1 .
Solving for x Solving for x , we get x = − 1 . Therefore, lo g 5 1 5 = − 1 .
Final Answer Thus, the value of the logarithmic expression is − 1 .
Examples
Logarithms are used in many real-world applications, such as measuring the magnitude of earthquakes on the Richter scale, determining the acidity or alkalinity (pH) of a solution, and modeling population growth or decay. In finance, logarithms are used to calculate the time it takes for an investment to double at a given interest rate. Understanding logarithms helps in making informed decisions in various fields.
