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Study Guides > MATH 1314: College Algebra

Key Concepts & Glossary

Key Equations

Definition of the logarithmic function For  x>0,b>0,b1\text{ } x>0,b>0,b\ne 1\\,y=logb(x)y={\mathrm{log}}_{b}\left(x\right)\\ if and only if  by=x\text{ }{b}^{y}=x\\.
Definition of the common logarithm For  x>0\text{ }x>0\\, y=log(x)y=\mathrm{log}\left(x\right)\\ if and only if  10y=x\text{ }{10}^{y}=x\\.
Definition of the natural logarithm For  x>0\text{ }x>0\\, y=ln(x)y=\mathrm{ln}\left(x\right)\\ if and only if  ey=x\text{ }{e}^{y}=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 their 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.
  • Real-world exponential problems with base 10 can be rewritten as a common logarithm and then evaluated using a calculator.
  • Natural logarithms can be evaluated using a calculator.

Glossary

common logarithm
the exponent to which 10 must be raised to get x; log10(x){\mathrm{log}}_{10}\left(x\right)\\ is written simply as log(x)\mathrm{log}\left(x\right)\\.
logarithm
the exponent to which b must be raised to get x; written y=logb(x)y={\mathrm{log}}_{b}\left(x\right)\\
natural logarithm
the exponent to which the number e must be raised to get x; loge(x){\mathrm{log}}_{e}\left(x\right)\\ is written as ln(x)\mathrm{ln}\left(x\right)\\.

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