`color{red}{x → log_b x = y}` if `b^y = x`
● This function, called the logarithmic function, is defined in `log_b : R^+ → R`
● As before if the base `b = 10,` we say it is common logarithms and if `b = e,` then we say it is natural logarithms.
Often natural logarithm is denoted by `ln.`
Graph of `log`
Properties :
1. `a^(log_(a) x) = x ; a ne 0 , ne 1 , x > 0`
2. `a^(log_(b) x ) = x^(log_b a) ; a , b > 0 , ne 1 , x > 0`
3. `log_a a =1, a > 0 , ne 1`
4. `log_a x = 1/(log_x a) ; x , a > 0 , ne 1`
5. `log_a x = (log_b x)/( log_b a) ; a , b > 0 , ne 1 , x > 0`
6. For `m, n > 0` and `a > 0,` `ne 1`, then
(i) `log_a (m * n) = log_a m + log_a n`
(ii) `log_a (m/n) = log_a m - log_a n`
(iii) `log_a (m^n) = n log_a m`
`color{blue}{"Theorem"}` : `color{green}{"The derivative of log" \ \ x \ \ w.r.t., x\ \ i s\ \ 1/x}`
`=>` `d/(dx) (log x) =1/x`
`color{red}{x → log_b x = y}` if `b^y = x`
● This function, called the logarithmic function, is defined in `log_b : R^+ → R`
● As before if the base `b = 10,` we say it is common logarithms and if `b = e,` then we say it is natural logarithms.
Often natural logarithm is denoted by `ln.`
Graph of `log`
Properties :
1. `a^(log_(a) x) = x ; a ne 0 , ne 1 , x > 0`
2. `a^(log_(b) x ) = x^(log_b a) ; a , b > 0 , ne 1 , x > 0`
3. `log_a a =1, a > 0 , ne 1`
4. `log_a x = 1/(log_x a) ; x , a > 0 , ne 1`
5. `log_a x = (log_b x)/( log_b a) ; a , b > 0 , ne 1 , x > 0`
6. For `m, n > 0` and `a > 0,` `ne 1`, then
(i) `log_a (m * n) = log_a m + log_a n`
(ii) `log_a (m/n) = log_a m - log_a n`
(iii) `log_a (m^n) = n log_a m`
`color{blue}{"Theorem"}` : `color{green}{"The derivative of log" \ \ x \ \ w.r.t., x\ \ i s\ \ 1/x}`
`=>` `d/(dx) (log x) =1/x`