Logarithms
When the unknown is an exponent, what number is it, and how do you calculate with numbers like that?
Where you are 11 inside
What this area is about
For a base b > 0 with b \ne 1 and a number x > 0, \log_b x is the exponent b must be raised to in order to give x. So \log_b x = y says exactly what b^y = x says, and y = \log_b x undoes y = b^x. Logarithms turn multiplying into adding, which is why they solve equations with the unknown in an exponent and measure quantities that span many powers of ten.
How these fit together
Definition and values. Definition of a logarithm turns a logarithm into a power statement and back. Evaluating logarithms finds the value, exactly or with the \log and \ln keys, and change of base reaches any other base from those two.
The laws. The laws of logarithms turn the logarithm of a product, quotient or power into a sum, difference or multiple.
Procedures. Graphing logarithmic functions reflects the exponential’s graph in the line y = x and moves it. Expanding and condensing uses the laws together. Matching bases and taking a logarithm of both sides solve exponential equations. Solving logarithmic equations drops any answer that puts a number that is not positive inside a logarithm.
Applications. Logarithmic scales such as pH and decibels, and semi-log plots, which straighten exponential data.
A path through it
The definition first, then evaluating and the laws. The laws open change of base, expanding and both ways of solving exponential equations; graphs come before solving logarithmic equations. Logarithmic scales lead into semi-log plots.
Next: fitting models by transforming data and, in calculus, derivatives of logarithms.