Log functions

Logarithmic functions are introduced as a significant type of function in mathematics.
Definition and notation:
- Suppose that ( x ) and ( b ) are positive real numbers, and ( b \neq 1 ).
- The logarithmic function base ( b ) is defined as ( y = \log_b{x} ).
- This relationship can be expressed in exponential form as ( b^y = x ).
Interpretation:
- Logarithm as an exponent: Think of a logarithm as the exponent to which the base must be raised to yield the argument.

Example 1: ( y = \log_2{8} )
- This can be stated in exponential form as ( 2^y = 8 ).
- Identifying ( y ): Since ( 2^3 = 8 ), thus ( y = 3 ).
- Conclusion: ( \log_2{8} = 3 ).
Example 2: ( y = \log_5{25} )
- Exponential form: ( 5^y = 25 ).
- Determining ( y ): As ( 5^2 = 25 ), therefore ( y = 2 ).
- Conclusion: ( \log_5{25} = 2 ).

Example 3: Writing logarithms as exponential statements
- Convert ( \log_8{64} = 2 ) to exponential:
- Exponential form: ( 8^2 = 64 ).
- Convert ( \log_{10}{\frac{1}{10000}} = -4 ) to exponential:
- Since there's no base mentioned, it is implicitly base 10:
- Exponential form: ( 10^{-4} = \frac{1}{10000} ).
- Convert ( \log_4{1} = 0 ) to exponential:
- Exponential form: ( 4^0 = 1 ), which is true by the definition of exponents.

Example 4: Converting exponential statements into logarithmic forms
- Convert ( 5^3 = 125 ) to logarithmic form:
- ( \log_5{125} = 3 ).
- Convert ( \left(\frac{1}{5}\right)^{-3} = 125 ) to logarithmic form:
- ( \log_{\frac{1}{5}}{125} = -3 ).
- Convert ( 10^9 = 1,000,000,000 ) to logarithmic form:
- Result: ( \log_{10}{1,000,000,000} = 9 ), typically written as ( \log{1,000,000,000} = 9 ) due to the implicit base of 10.

Common Logarithmic Function:
- This is the logarithm with base 10, written as ( y = \log_{10}{x} ) or simply ( y = \log{x} ).
Natural Logarithmic Function:
- This is the logarithm with base e, denoted as ( y = \log_e{x} ) or ( y = \ln{x} ), which stands for "log natural".

Example 5: Evaluating logarithms
- Evaluate ( y = \log{100000000} ):
- Exponential form: ( 10^y = 100000000 ).
- Since ( 100000000 = 10^8 ), we have ( y = 8 ).
- Evaluate ( y = \ln{\left(\frac{1}{e^3}\right)} ):
- Exponential form: ( e^y = \frac{1}{e^3} ).
- ( \frac{1}{e^3} = e^{-3} ) so ( y = -3 ).

Common Logarithm of 29416:
- The argument lies between ( 10^4 ) and ( 10^5 ).
- Use calculator: ( \log{29416} \approx 4.4686 ) (as expected, between 4 and 5).
Common Logarithm of 0.042:
- Engage approximation, as it falls between ( 10^{-2} ) and ( 10^{-1} ).
- Use calculator: ( \log{0.042} \approx -1.3768 ) (correctly between -2 and -1).
Natural Logarithm of 340:
- Use calculator: ( \ln{340} \approx 5.8289 ).
Natural Logarithm of 2π:
- Use calculator: ( \ln{2\pi} \approx 1.8379 ).

Property 1: ( \log_b{1} = 0 )
- Reason: Any non-zero base raised to the zero power equals one.
Property 2: ( \log_b{b} = 1 )
- Reason: Because ( b^1 = b ).
Property 3: ( \log_b{b^x} = x )
- Reason: As it translates to ( b^x = b^x ) (equal bases yields the exponent).
Property 4: ( b^{\log_b{x}} = x )
- Reason: Composing a base with its logarithm retrieves the original value.