Vocabulary
Logarithm of y with base b- Let b and y be positive numbers with b≠1. The logarithm of y with base b, denoted logᵦy and read "log base b of y," is defined as follows: logᵦy=x if and only if b^x =y.
Common logarithm- A logarithm with base 10. It is denoted by log₁₀ or simply by log.
Natural logarithm- A logarithm with base e. It can be denoted logₑ, but is more often denoted by ln.
Common logarithm- A logarithm with base 10. It is denoted by log₁₀ or simply by log.
Natural logarithm- A logarithm with base e. It can be denoted logₑ, but is more often denoted by ln.
Essential Information
What is the value of x in 2^x=6? The x value can be defined by writing a logarithm and write x=log₂6.
Definition of Logarithm with Base b
Let b and y be positive numbers with b≠1. The logarithm of y with base b, denoted logᵦy and read "log base b of y," is defined as follows: logᵦy=x if and only if b^x =y.
logᵦy=x and b^x =y are equivalent. The first is in logarithmic form and the second is in exponential form.
Definition of Logarithm with Base b
Let b and y be positive numbers with b≠1. The logarithm of y with base b, denoted logᵦy and read "log base b of y," is defined as follows: logᵦy=x if and only if b^x =y.
logᵦy=x and b^x =y are equivalent. The first is in logarithmic form and the second is in exponential form.
b and c illustrate two special logarithms. Let b be a positive real number such that b≠1.
Special Logarithms
A common logarithm is a logarithm with base 10. It can be denoted by log₁₀ or simply by log. A natural logarithm is a logarithm with base e. It can be denoted by logₑ, but is more often denoted by ln.
- logᵦ1=o because b⁰=1.
- logᵦb=1 because b¹=b.
Special Logarithms
A common logarithm is a logarithm with base 10. It can be denoted by log₁₀ or simply by log. A natural logarithm is a logarithm with base e. It can be denoted by logₑ, but is more often denoted by ln.
- Common logarithm: log₁₀x=log x
- Natural logarithm logₑx=ln x
Inverse Functions
By the definition of a logarithm, it follows that the logarithmic function g(x)=logᵦx is the inverse of the exponential function f(x)=b^x. This means that:
g(f(x)) = logᵦb^x = x and f(g(x)) + b^(logᵦx =x)
By the definition of a logarithm, it follows that the logarithmic function g(x)=logᵦx is the inverse of the exponential function f(x)=b^x. This means that:
g(f(x)) = logᵦb^x = x and f(g(x)) + b^(logᵦx =x)
Graphing Logarithmic Functions
Inverse relations can be used between exponential and logarithmic functions to graph logarithmic functions.
Parent Graphs for Logarithmic Functions
Because f(x) = logᵦx and g(x) + b^x are inverse functions, the graph of f(x) = logᵦx is the reflection of the graph of g(x) = b^x in the line y=x.
Inverse relations can be used between exponential and logarithmic functions to graph logarithmic functions.
Parent Graphs for Logarithmic Functions
Because f(x) = logᵦx and g(x) + b^x are inverse functions, the graph of f(x) = logᵦx is the reflection of the graph of g(x) = b^x in the line y=x.
TranslationsLogarithmic functions of the form y=logᵦ(x - h) + k can be graphed by translating the graph of the parent function y= logᵦx.