Solving the Equation
In this article, we will solve the equation:
$\log_2(100x)+\log_2(10x)=14+\log_2\left(\frac{1}{x}\right)$
Step 1: Simplify the Left-Hand Side
Using the property of logarithms, we can rewrite the left-hand side as:
$\log_2(100x)+\log_2(10x)=\log_2(1000x^2)$
Step 2: Simplify the Right-Hand Side
Using the property of logarithms, we can rewrite the right-hand side as:
$14+\log_2\left(\frac{1}{x}\right)=14-\log_2(x)$
Step 3: Equate the Two Expressions
Equating the two expressions, we get:
$\log_2(1000x^2)=14-\log_2(x)$
Step 4: Solve for x
To solve for x, we can rewrite the equation as:
$\log_2(x^2)=14-\log_2(1000)$
Using the property of logarithms, we can rewrite the equation as:
$\log_2(x^2)=\log_2\left(\frac{2^{14}}{1000}\right)$
Taking the exponential of both sides, we get:
$x^2=\frac{2^{14}}{1000}$
Solving for x, we get:
$x=\sqrt{\frac{2^{14}}{1000}}=\frac{2^7}{10}$
Therefore, the solution to the equation is:
$x=\frac{128}{10}=\boxed{12.8}$