For [latex]\\text{ } x>0,b>0,b\\ne 1[\/latex], [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] if and only if [latex]\\text{ }{b}^{y}=x[\/latex].<\/td>\n<\/tr>\n
\n
Definition of the common logarithm<\/td>\n
For [latex]\\text{ }x>0[\/latex], [latex]y=\\mathrm{log}\\left(x\\right)[\/latex] if and only if [latex]\\text{ }{10}^{y}=x[\/latex].<\/td>\n<\/tr>\n
\n
Definition of the natural logarithm<\/td>\n
For [latex]\\text{ }x>0[\/latex], [latex]y=\\mathrm{ln}\\left(x\\right)[\/latex] if and only if [latex]\\text{ }{e}^{y}=x[\/latex].<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
Key Concepts<\/h2>\n
\n \t
The inverse of an exponential function is a logarithmic function, and the inverse of a logarithmic function is an exponential function.<\/li>\n \t
Logarithmic equations can be written in an equivalent exponential form using the definition of a logarithm.<\/li>\n \t
Exponential equations can be written in an equivalent logarithmic form using the definition of a logarithm.<\/li>\n \t
Logarithmic functions with base b<\/em> can be evaluated mentally using previous knowledge of powers of b<\/em>.<\/li>\n \t
Common logarithms can be evaluated mentally using previous knowledge of powers of 10.<\/li>\n \t
When common logarithms cannot be evaluated mentally, a calculator can be used.<\/li>\n \t
Natural logarithms can be evaluated using a calculator.<\/li>\n<\/ul>\n
Glossary<\/h2>\n
\n \t
common logarithm<\/strong><\/dt>\n \t
the exponent to which 10 must be raised to get x<\/em>; [latex]{\\mathrm{log}}_{10}\\left(x\\right)[\/latex] is written simply as [latex]\\mathrm{log}\\left(x\\right)[\/latex]<\/dd>\n<\/dl>\n
\n \t
logarithm<\/strong><\/dt>\n \t
the exponent to which b<\/em> must be raised to get x<\/em>; written [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/dd>\n<\/dl>\n
\n \t
natural logarithm<\/strong><\/dt>\n \t
the exponent to which the number e<\/em> must be raised to get x<\/em>; [latex]{\\mathrm{log}}_{e}\\left(x\\right)[\/latex] is written as [latex]\\mathrm{ln}\\left(x\\right)[\/latex]<\/dd>\n<\/dl>\n\n","rendered":"
Key Equations<\/h2>\n
\n\n
\n
Definition of the logarithmic function<\/td>\n
For [latex]\\text{ } x>0,b>0,b\\ne 1[\/latex], [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex] if and only if [latex]\\text{ }{b}^{y}=x[\/latex].<\/td>\n<\/tr>\n
\n
Definition of the common logarithm<\/td>\n
For [latex]\\text{ }x>0[\/latex], [latex]y=\\mathrm{log}\\left(x\\right)[\/latex] if and only if [latex]\\text{ }{10}^{y}=x[\/latex].<\/td>\n<\/tr>\n
\n
Definition of the natural logarithm<\/td>\n
For [latex]\\text{ }x>0[\/latex], [latex]y=\\mathrm{ln}\\left(x\\right)[\/latex] if and only if [latex]\\text{ }{e}^{y}=x[\/latex].<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
Key Concepts<\/h2>\n
\n
The inverse of an exponential function is a logarithmic function, and the inverse of a logarithmic function is an exponential function.<\/li>\n
Logarithmic equations can be written in an equivalent exponential form using the definition of a logarithm.<\/li>\n
Exponential equations can be written in an equivalent logarithmic form using the definition of a logarithm.<\/li>\n
Logarithmic functions with base b<\/em> can be evaluated mentally using previous knowledge of powers of b<\/em>.<\/li>\n
Common logarithms can be evaluated mentally using previous knowledge of powers of 10.<\/li>\n
When common logarithms cannot be evaluated mentally, a calculator can be used.<\/li>\n
Natural logarithms can be evaluated using a calculator.<\/li>\n<\/ul>\n
Glossary<\/h2>\n
\n
common logarithm<\/strong><\/dt>\n
the exponent to which 10 must be raised to get x<\/em>; [latex]{\\mathrm{log}}_{10}\\left(x\\right)[\/latex] is written simply as [latex]\\mathrm{log}\\left(x\\right)[\/latex]<\/dd>\n<\/dl>\n
\n
logarithm<\/strong><\/dt>\n
the exponent to which b<\/em> must be raised to get x<\/em>; written [latex]y={\\mathrm{log}}_{b}\\left(x\\right)[\/latex]<\/dd>\n<\/dl>\n
\n
natural logarithm<\/strong><\/dt>\n
the exponent to which the number e<\/em> must be raised to get x<\/em>; [latex]{\\mathrm{log}}_{e}\\left(x\\right)[\/latex] is written as [latex]\\mathrm{ln}\\left(x\\right)[\/latex]<\/dd>\n<\/dl>\n\n\t\t\t \n\t\t\t