Understanding logarithms is essential as they serve as the inverse of exponential functions. This relationship allows us to evaluate logarithmic expressions without a calculator. For instance, consider the expression log2(∛2). By recognizing that the cube root of 2 can be expressed as an exponent, we rewrite it as log2(21/3). Utilizing the inverse property of logarithms, where logb(bx) = x, we find that log2(21/3) = 1/3.

Another important property is that logb(b) = 1. This means that the logarithm of a base to itself always equals one. For example, log2(2) = 1 and log10(10) = 1. Similarly, the logarithm of 1 in any base is always zero, as logb(1) = 0. This is because any number raised to the power of zero equals one, such as 20 = 1.

To further illustrate these properties, let’s evaluate a few examples. For log2(∛2), we rewrite it as log2(21/3), which simplifies to 1/3. For the natural logarithm of 1, ln(1), we recognize it as loge(1), which equals 0. In the case of log(10), this is equivalent to log10(10), yielding 1.

Lastly, consider log5(1/5). We can express 1/5 as 5-1, allowing us to rewrite the logarithm as log5(5-1). Applying the inverse property, we find that this simplifies to -1.

By mastering these properties and techniques, evaluating logarithmic expressions becomes straightforward, enabling you to solve problems efficiently without the need for a calculator.