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, leading to the solution \( x = 3 \).
For more complex equations like \( 2^x = 216 \), instead of multiplying 2 repeatedly, we can utilize logarithms. The logarithm is the inverse operation of exponentiation. To isolate \( x \), we take the logarithm of both sides, specifically using the same base as the exponential. Thus, we apply \( \log_2(2^x) = \log_2(216) \). Since the bases match, this simplifies to \( x = \log_2(216) \).
The logarithmic expression \( \log_2(216) \) represents the power to which the base 2 must be raised to yield 216. This logarithmic form is equivalent to the original exponential equation \( 2^x = 216 \). Understanding how to convert between exponential and logarithmic forms is crucial. For instance, converting \( 3^x = 81 \) to logarithmic form results in \( \log_3(81) = x \). Conversely, if given \( x = \log_4(64) \), the equivalent exponential form is \( 4^x = 64 \).
To further illustrate, consider \( x = \log_5(800) \). The conversion to exponential form gives \( 5^x = 800 \). Similarly, for \( \log_2(16) = 4 \), the exponential form is \( 2^4 = 16 \), confirming the accuracy of the conversion. Lastly, for \( 10^x = 45100 \), the logarithmic form is \( x = \log_{10}(45100) \), which can be simplified to just \( \log(45100) \) since base 10 logarithms are commonly referred to as common logs.
Understanding these conversions and the properties of logarithms will greatly enhance your ability to solve equations involving exponents and logarithms effectively.
