When solving polynomial equations, such as \(x^3 = 216\), the goal is to isolate \(x\) by performing the reverse operation. In this case, taking the cube root of both sides yields \(x = \sqrt[3]{216}\). However, when the variable \(x\) is in the exponent, as in \(2^x = 8\), we need to determine how many times \(2\) must be multiplied by itself to equal \(8\). The answer is \(x = 3\). For more complex equations like \(2^x = 216\), instead of multiplying \(2\) repeatedly, we can use logarithms to simplify the process.
The logarithm is the inverse operation of exponentiation. To isolate \(x\) in the equation \(2^x = 216\), we take the logarithm of both sides, specifically using base \(2\) to match the base of the exponential. This gives us \(x = \log_2{216}\). In this logarithmic form, the expression indicates the power to which the base \(2\) must be raised to yield \(216\). This relationship is equivalent to the original exponential equation.
To convert between exponential and logarithmic forms, always start with the base of the exponent. For example, converting \(3^x = 81\) to logarithmic form results in \(x = \log_3{81}\). Conversely, to change \(x = \log_4{64}\) into exponential form, we recognize that \(4^x = 64\). This method of conversion is consistent: begin with the base, move to the other side of the equation, and then return to the starting point.
For further practice, consider \(x = \log_5{800}\). The equivalent exponential form is \(5^x = 800\). Similarly, for \( \log_2{16} = 4\), the exponential form is \(2^4 = 16\), confirming the accuracy of the conversion. Lastly, the equation \(10^x = 45100\) can be expressed as \(x = \log_{10}{45100}\), which is often referred to as the common logarithm, denoted simply as \(\log{45100}\). This logarithm is frequently used and has a dedicated button on calculators for easy evaluation.
Understanding these conversions and the properties of logarithms will greatly enhance your ability to solve equations involving exponents efficiently.
