Exponentiation determines the result of repeatedly multiplying a value (the base) by itself a specific number of times (that is, the exponent), resulting in the “power”.
Logarithms can “undo” this operation by discovering the exponent to which a base must be raised to obtain some specific value.
Often, “log” is said instead of logarithm, due to its symbol: $\log$
Now, a typical logarithm looks as follows:
$$ \log_{\text{base}}(\text{anti-logarithm}) = \text{exponent} $$The antilog, or antilogarithm, is the “power” in a typical exponentiation expression (it is the result of the exponentiation). The base is the value such that $\text{base}^{\text{exponent}} = \text{antilog}$.
Now, you may see that the base is omitted. In such cases, it is assumed that the base value is actually $10$ (since, numbers are in base-10).
A special type of log known as the Natural Logarithm is a logarithm such that the base is the Natural Number $e$ (aka Euler’s Constant). This is expressed as:
$$ \log_{e}(x) = \ln(x) $$This is spoken as “The natural log of $x$”
Let’s look at some examples:
$$ \log_{10}(100)=2 $$This is because the base, $10$ must be raised to the power of $2$ in order to have a result of $100$. But, not every logarithm is so clean. More than likely, your resultant exponent will not be a whole number, such as:
$$ \log_{5}(100)\approx2.86135... $$Logarithms have some unique properties. These equalities enable great simplifications when working with them. Use these as a reference when working on problems to allow immense simplifications.
$$ \begin{aligned} \log_b{(M \cdot N)} &= \log_b{M} + \log_b{N} \\ \log_b{\frac{M}{N}} &= \log_b{M}-\log_b{N} \\ \log_b({M^k}) &= k \cdot \log_b{M} \\ \log_b{1} &= 0 \\ \log_b{b} &= 1 \\ \log_b(b^k) &= k \\ b^{log_b{k}} &= k \\ \end{aligned} $$