11. \( \log _{9} 1=x \)

To solve the equation \( \log_{9} 1 = x \), remember that the logarithm tells us the power to which the base must be raised to produce a given number. Since any number raised to the power of 0 equals 1, we have \( 9^0 = 1 \). Therefore, \( x = 0 \) is the solution! Now, you might wonder why logarithms are useful. They’re not just academic; they’re used in everything from calculating pH levels in chemistry to determining the decibel level of sound. Logarithms can simplify complex calculations and are essential in algorithms that underpin search engines and data compression techniques.