Summary: Logarithmic Functions

Key Equations

Definition of the logarithmic function For  x>0,b>0,b1, y=logb(x) if and only if  by=x.
Definition of the common logarithm For  x>0, y=log(x) if and only if  10y=x.
Definition of the natural logarithm For  x>0, y=ln(x) if and only if  ey=x.

Key Concepts

  • The inverse of an exponential function is a logarithmic function, and the inverse of a logarithmic function is an exponential function.
  • Logarithmic equations can be written in an equivalent exponential form using the definition of a logarithm.
  • Exponential equations can be written in an equivalent logarithmic form using the definition of a logarithm.
  • Logarithmic functions with base b can be evaluated mentally using previous knowledge of powers of b.
  • Common logarithms can be evaluated mentally using previous knowledge of powers of 10.
  • When common logarithms cannot be evaluated mentally, a calculator can be used.
  • Natural logarithms can be evaluated using a calculator.

Glossary

common logarithm
the exponent to which 10 must be raised to get x; log10(x) is written simply as log(x)
logarithm
the exponent to which b must be raised to get x; written y=logb(x)
natural logarithm
the exponent to which the number e must be raised to get x; loge(x) is written as ln(x)