Solve for $x$.
$\log _x \frac{1}{100}=2$
Simplify your answer as much as possible.
Answer (1)
Rewrite the logarithmic equation in exponential form: x 2 = 100 1 .
Take the square root of both sides: x = ± 10 1 .
Discard the negative solution since the base of a logarithm must be positive: x = 10 1 .
The solution is 10 1 .
Explanation
Understanding the Problem We are given the equation lo g x 100 1 = 2 . Our goal is to solve for x , which represents the base of the logarithm.
Converting to Exponential Form To solve for x , we need to rewrite the logarithmic equation in its equivalent exponential form. Recall that lo g b a = c is equivalent to b c = a . Applying this to our equation, we get: x 2 = 100 1
Taking the Square Root Now, we take the square root of both sides of the equation to isolate x : x 2 = 100 1 This gives us: x = ± 10 1
Considering the Base of Logarithm However, the base of a logarithm must be positive. Therefore, we discard the negative solution. This leaves us with: x = 10 1
Verification To verify our solution, we substitute x = 10 1 back into the original equation: lo g 10 1 100 1 = 2 Since ( 10 1 ) 2 = 100 1 , our solution is correct.
Final Answer Therefore, the solution to the equation lo g x 100 1 = 2 is: x = 10 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 of a solution using pH, and modeling population growth or radioactive decay. In finance, logarithms are used to calculate the time it takes for an investment to double at a given interest rate. Understanding how to solve logarithmic equations is essential for making informed decisions in various fields.
