Mathematics LOGARITHMIC INEQUALITIES
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LOGARITHMIC INEQUALITIES

Log Inequalities :

(I) For `a > 1`, If `log_a x > log_a y` , then `=> x > y` that is if base is greater than unity the inequality remains unchanged when log is removed.

(2) For `0< a <1` lf `log_ax > log_by` , then `=> x < y` that is if base is less than unity then inequality sign is reversed when log is removed.

(3) lf `a > 1 . log_ax < P => x < a^P`

(4) lf `a > 1 . log_ax > P => x > a^P`

(5) lf `a < 1 . log_ax < P => x > a^P`

(6) lf `a < 1 . log_ax > P => x < a^P`

`=>` If base is less than unity then value of `log x` decreases as `x`-increases.

`=>` Jf base is greater than unity then value of `log x` increases as `x` increases.

logarithmic lnequations

When, we solve logarithmic inequations


`{tt ((log_a f(x) > log_a g(x)),(a > 1)) => {tt ((g(x) >0),(a > 1),(f(x) > g(x))) `

`{tt ((log_a f(x) > log_a g(x)),(0 < a < 1)) => {tt ((g(x) >0),(0 < a < 1),(f(x) < g(x))) `


`text(Some Standard Forms to Solve Logarithmic lnequations :)`

`text(Form 1)` Inequations of the form

`text(Form) text( ) text(Collection of systems)`
`(a) log_(g(x)) f(x) > 0text( ) ⇔ text( ){tt[(f{x} > 1),(g{x} > 1)] {tt[(0 ,< , f{x} <1),(0 ,< , g{x} <1)] `
`(b) log_(g(x)) f(x) >= 0text( ) ⇔ text( ){tt[(f{x} >= 1),(g{x} > 1)] {tt[(0 ,< , f{x} <= 1),(0 ,< , g{x} <1)] `
`(c) log_(g(x)) f(x) < 0text( ) ⇔ text( ){tt[(f{x} > 1),(0 < g{x} > 1)] {tt[(0 ,< , f{x} <1),( quad, quad, g{x} <1)] `
`(d) log_(g(x)) f(x) >= 0text( ) ⇔ text( ){tt[(f{x} >= 1),(0 < g{x} < 1)] {tt[(0 ,< , f{x} =< 1),( quad, quad, g{x} <1)] `

`text(Form 2)` Inequations of the form

`text(Form) text( ) text(Collection of systems)`
`(a) log_(phi(x))f(x)>log_(phi(x))g(x)text( )⇔text( ) {tt[(f{x} >= g{x}),(g{x}>0),(phi{x} > 1)] {tt[(f{x} < g{x}),(g{x}>0),(0 < phi{x} > 1)]`
`(b) log_(phi(x))f(x)>=log_(phi(x))g(x)text( )⇔text( ) {tt[(f{x} > g{x}),(g{x}>0),(phi{x} > 1)] {tt[(f{x} <= g{x}),(g{x}>0),(0 < phi{x} > 1)]`
`(c) log_(phi(x))f(x)>log_(phi(x))g(x)text( )⇔text( ) {tt[(f{x} > g{x}),(g{x}>0),(phi{x} > 1)] {tt[(f{x} > g{x}),(g{x}>0),(0 < phi{x} > 1)]`
`(d) log_(phi(x))f(x)>=log_(phi(x))g(x)text( )⇔text( ) {tt[(f{x} <= g{x}),(g{x}>0),(phi{x} > 1)] {tt[(f{x} >= g{x}),(g{x}>0),(0 < phi{x} > 1)]`
 
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