Solve for x.
$\log _2 x=1$
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The solution is $x=$ $\square$ .
(Simplify your answer. Type an integer or a fraction.)
B. The solution is not a real number.
Answer (2)
Rewrite the logarithmic equation in exponential form: 2 1 = x .
Simplify the equation.
Find the value of x.
The solution is 2 .
Explanation
Understanding the Problem We are given the equation lo g 2 x = 1 . Our goal is to solve for x .
Converting to Exponential Form To solve the logarithmic equation, we need to rewrite it in exponential form. Recall that lo g b a = c is equivalent to b c = a . In our case, we have b = 2 , a = x , and c = 1 .
Applying the Definition of Logarithm Rewriting the equation lo g 2 x = 1 in exponential form, we get 2 1 = x .
Solving for x Simplifying the equation 2 1 = x , we find that x = 2 .
Final Answer Therefore, the solution is x = 2 .
Examples
Logarithms are used to solve many real-world problems, such as calculating the magnitude of earthquakes on the Richter scale or determining the pH of a chemical solution. For example, if we know the magnitude of an earthquake is 3 on the Richter scale, we can use logarithms to determine the amplitude of the seismic waves. Similarly, in finance, logarithms are used to calculate the time it takes for an investment to double at a certain interest rate. Understanding logarithms helps us to quantify and analyze phenomena in various scientific and practical contexts.
The solution to the equation lo g 2 x = 1 is x = 2 .
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