Applications to Logarithms and Geometry 1

The logarithm of a positive real number x with respect to base b[nb 1] is the exponent by which b must be raised to yield x. In other words, the logarithm of x to base b is the solution y to the equation[2]

{\displaystyle b^{y}=x.} {\displaystyle b^{y}=x.}
The logarithm is denoted "logb x" (pronounced as "the logarithm of x to base b" or "the base-b logarithm of x" or (most commonly) "the log, base b, of x").

In the equation y = logb x, the value y is the answer to the question "To what power must b be raised, in order to yield x?".

Examples
log2 16 = 4 , since 24 = 2?×2?×?2?×?2 = 16.
Logarithms can also be negative: {\displaystyle \quad \log _{2}\!{\frac {1}{2}}=-1\quad } {\displaystyle \quad \log _{2}\!{\frac {1}{2}}=-1\quad } since {\displaystyle \quad 2^{-1}={\frac {1}{2^{1}}}={\frac {1}{2}}.} {\displaystyle \quad 2^{-1}={\frac {1}{2^{1}}}={\frac {1}{2}}.}
log10150 is approximately 2.176, which lies between 2 and 3, just as 150 lies between 102 = 100 and 103 = 1000.
For any base b, logb b = 1 and logb 1 = 0, since b1 = b and b0 = 1, respectively.