The number of real solution of the equation \(\left(1 / 27^{\mathrm{x}}\right)+\left(1 / 12^{\mathrm{x}}\right)=\left(1 / 2.8^{\mathrm{x}}\right)\) (a) 1 (b) 2 (b) 3 (d) 0

Logarithms are incredibly useful in mathematics for simplifying expressions involving exponents. They are the inverse operation of exponentiation and are used to solve equations where the variable is an exponent.

What does that mean? Well, if we have an equation like \( a^x = b \), taking the logarithm of both sides—most often the natural logarithm (ln) or the base ten logarithm (log)—allows us to solve for the unknown exponent \( x \). That's because the basic property of logarithms states that \( \text{log}_a(b) = x \) if and only if \( a^x = b \). This principle is pivotal for solving exponential equations, which we will see in the context of our exercise.

Exponential functions, on the other hand, are functions of the form \( f(x) = a^x \), where \( a \) is a positive constant, and they show how quantities grow or decay at rates proportional to their size. They pop up in various scientific fields, including physics, biology, and economics.

In our equation-solving exercise, exponential functions are evident on each side of the initial equation. They can be tricky to work with because the variable we're solving for is the exponent. This is where our understanding of logarithms becomes essential. By using logarithms, we can 'bring the exponent down' and solve for the variable \( x \). Unfortunately, in our exercise, it appears that after applying logarithms, we encounter an inconsistency that leads to the conclusion there are no real solutions.

Equation solving is all about finding the value(s) of the variable(s) that make a mathematical statement true. The problem-solving process typically involves simplifying expressions and isolating variables. In the case of our exercise, we attempted to use properties of logarithms to simplify and solve an exponential equation.

The ability to recognize when to apply certain strategies, like using logarithms to deal with exponentials, is crucial. However, when solving equations, it's also important to remember to check all possible solutions, including the special case when the variable equals zero, known as the 'zero exponent rule', where any nonzero number raised to the power of zero equals one. In our example, this was the final step, which confirmed that there were, indeed, zero real solutions to the equation.