Determine whether the following statements are true and give an explanation or counterexample.


$2=10^{{\log }_{10}2}$

Key Concepts

Logarithmic Identity

The logarithmic identity states that for any positive number 'a' and 'b', the equation a = b^log_b(a) holds true. This means that raising the base 'b' to the logarithm of 'a' with base 'b' will yield 'a'. This identity is fundamental in understanding how logarithms relate to exponentiation and is crucial for evaluating expressions involving logarithms.

Properties of Logarithms

Logarithms have several key properties that simplify calculations, such as the product, quotient, and power rules. For instance, log_b(xy) = log_b(x) + log_b(y) and log_b(x/y) = log_b(x) - log_b(y). Understanding these properties allows for the manipulation of logarithmic expressions, which is essential for verifying the truth of statements involving logarithms.

Base of a Logarithm

The base of a logarithm is the number that is raised to a power to obtain a given number. In the expression log_b(a), 'b' is the base. Different bases can yield different results, and it is important to recognize that the base must be positive and not equal to one. This concept is critical when evaluating logarithmic expressions and understanding their implications in equations.