= Logarithm
{title2=$\log_b$}
{wiki}
For a positive real base $b\ne1$, the <logarithm> is the <inverse function> of the <exponential function> $t\mapsto b^t$:
$$
\log_b x=t\quad\Longleftrightarrow\quad b^t=x,\qquad x>0.
$$
It asks which exponent produces a given positive number. Thus $\log_2 8=3$ and $\log_2(1/8)=-3$. The base restrictions matter: $b=1$ would produce only one value, while a negative base does not define $b^t$ as a real-valued <function> for every real exponent.
The useful transformation is
$$
\log_b(xy)=\log_bx+\log_by,\qquad \log_b(x/y)=\log_bx-\log_by.
$$
For example, $8\cdot16=128$ corresponds to $3+4=7$ in base two. Products become sums and ratios become differences. The <logarithm> does not generally convert a sum into a sum of logarithms. Base conversion uses the <natural logarithm>: $\log_bx=\ln x/\ln b$. It also gives $d(\log_bx)/dx=1/(x\ln b)$, positive when $b>1$ and negative when $0<b<1$.
Historically, <John Napier>'s 1614 Mirifici logarithmorum canonis descriptio presented logarithmic tables and their use, documented in https://library.si.edu/digital-library/book/mirificilogarit00napi[the Smithsonian Libraries' digitized original]. Tables made multiplication-heavy calculations more manageable by replacing products with addition and a lookup. The 1619 Constructio title page below belongs to a later publication from the same work on <logarithms>. These historical publications preceded the modern <inverse function> and <integral> presentation used here.
\Image[https://upload.wikimedia.org/wikipedia/commons/3/3b/Napier's_Constructio_Latin_title_page.jpg]
{title=Title page of Napier's 1619 Mirifici logarithmorum canonis constructio}
{height=650}
{source=https://commons.wikimedia.org/wiki/File:Napier's_Constructio_Latin_title_page.jpg}
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