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”.
\color{cyan}{\textnormal{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 \color{cyan}{\textnormal{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 \color{cyan}{\textnormal{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.
\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
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