91Ó°ÊÓ

A logarithm is essentially a mathematical way to ask the question: 'To what exponent must a certain base be raised, to yield a specific number?' The logarithmic form of an equation is written as \(\text{log}_{b}(y) = x\), where \(b\) is the base, \(y\) is the result, and \(x\) is the exponent. This tells us that if we raise the base \(b\) to the power of \(x\), we will get \(y\). In our exercise, we start with the logarithmic equation \(\text{log}_{x}(18) = 2\). This asks the question: 'To what exponent must \(x\) be raised to produce 18?' Knowing how to read and understand equations in logarithmic form is the first step towards solving them.

To solve a logarithmic equation, it often helps to rewrite it in its exponential form. This transformation can make the equation much easier to handle. For our example, we convert \(\text{log}_{x}(18) = 2\) into an exponential form by interpreting it as \(x^2 = 18\). The reason this works is that we know, from the definition of logarithms, \(b^x = y\). This step essentially reverses the logarithmic function, making it simpler to work with.

After converting the logarithmic equation to exponential form, our next step is to solve for the unknown variable, \(x\). So, we start with the exponential equation \(x^2 = 18\). To find \(x\), we need to take the square root of both sides. Remember that taking the square root of a number yields both positive and negative solutions: \(x = \text{√}18\ \text{or} \ x = -\text{√}18\). However, because the base of a logarithm must be a positive number greater than 1, we discard the negative solution, leaving us with \(x = \text{√}18\). Simplifying further, we recognize that 18 can be expressed as \(9 \times 2\), so we simplify \(\text{√}18 = \text{√}(9 \times 2) = 3\text{√}2\). Therefore, our exact solution is \(x=3\text{√}2\).