In this section we'll learn about the
function logarithm. If you come across a
function like this then (log) means
logarithm function; where a is argument,
b is base and c is value of logarithm of a
to the base b and we can write a is
equal to b raised to power c. Where a
can be any positive value greater than
zero and b can be any positive value
greater than zero except 1. The value
of b cannot be 1 because log of a to
the base b can be written as log a /
log b to the base c and also log of 1
with any base is zero. So if b is equal
to 1 then log of b will be zero and
denominator will be 0 and if denominator
is zero function becomes not defined.
Hence base can not be 1. Whereas c can
be any real number R. Let's have some
example: if log of X to the base 3 is 2
then X will be 3 raised to power 2 that
is 9.
Similarly if log of 2x minus 1 to the
base half is equal to root 3 then 2x
minus 1 is equal to half to the power
root 3. If log of mod of X minus 1 to
base 3 is equal to 2 then mod of X minus
1 is 3 raised to power 2 that is 9. So,
mod of X minus 1 can be plus 9 or minus
9. So, X can be 10 or minus 8. Now let's
look at some important formulae. Now if
base b has some power k then we can
write 1 by k multiplied by logarithmic
function. Consider the example: here base
is 9. So, 9 can be written as 3 square.
Now, 2 can be taken out and it can be
written as half times of logarithmic
function. If any number is raised to the log
of X and if it has same base as the
number then we can directly write it as
X. For example: 2 raised to the power of
log of 7 to the base 2 will be 7. Log of
a plus log of b is equal to log of a
into b. If base is not written then we
can assume that the base is 10. Log of a
minus log of b will be log of a by b.
Log of  a raised to power n will be n log
of a. Log of a to the base b can be
written as log of a by log of b to the any 
 base c. Log to the base e is called
natural logarithm. So, log of X to the
base e will be natural logarithm of X or
ln of X. Value of e is roughly equal to 2.7. ln of 2 is nothing but the
natural logarithm of 2. The values a
and b can be interchanged by taking the
reciprocal of the logarithmic function.
Logarithm of 1 to the any base is always
0. if arguments and base have same values
then log will 1 for example natural log
of e will be 1 because natural log of e
is log of e to the base e and e e both
arguments and base have same values
hence answer will be 1. if log of a is
greater than log of c to the base b then
a is greater than C if b belongs to 1 to
infinity but a is less than C if b is
fractional means it belongs to 0 to 1
you can verify this result from the
graph on the right side for example if
log of X is less than log of 3 and base
is 1/2 since half lies between
zero to one that is friction hence X
will be greater than three and
inequality will be reversed consider
same example but in this case log of X
is greater than log of three here also
the inequality will be reversed and X
will be less than three
since half lies between 0 to 1 and we
know that argument can not be negative
it is always greater than zero hence the
overall solution will be x from 0 to 3
if log of X minus 1 is greater than or
equal to 2x plus 3 here
base is 3 which is greater than 1 hence
inequality will not be reversed here so
X minus 1 will be greater than or equal
to 2x plus 3 and we get X less than or
equal to minus 4 we also know that the
arguments of the logarithm function must
be positive hence X minus 1 must be
greater than 0 and 2x plus 3 must be
greater than 0 hence we get another two
equation plot all the three solution on
the number line here there is no
overlapping between all the three
solutions hence answer will be null set
means no solution how to solve this
expression log of mod of X to the base
root of 5 is greater than or equal to 2
mod of X will be greater than or equal
to root of 5 raised to power 2 hence mod
of X will be greater than or equal to 5
and in previous lessons we have already
learned that if mod of X is greater than
or equal to 5 then X will be either
greater than or equal to 5 or it will be
less than or equal to minus 5 let's plot
the solution on the number line so the
solution will be X from minus infinity
to minus 5 Union 5 to infinity consider
another example here you can notice that
the base is fractional and it is between
0 to 1 so the inequality will be
reversed here
hence mod of 2x minus 1 will be less
than or equal to 3 by 4 to the power 3
which gives mod of 2x minus 1 less than
or equal to 27 by 64 and we have already
learned in previous lessons that if mod
of X is less than or equal to any number
then X will lie between minus times of that
number to plus times of that number now
solving this expression will give the
values of X and answer and it is left
for you to solve the expression that's
all for this section thanks for watching :)
