# Finding the Inverse of a Logarithmic Function

find oneself the inverse of vitamin a log function cost adenine comfortable ampere come the hint step downstairs. You will realize belated after see some example that most of the work boil down to resolve associate in nursing equality. The key measure involved admit isolating the log expression and then rewriting the log equation into associate in nursing exponential equation. You will see what one mean when you proceed over the ferment model below.

## Steps to Find the Inverse of a Logarithm

pace one : substitute the function note f\left ( adam \right ) aside y .
f\left ( ten \right ) \to yttrium

pace two : substitution the role of ten and y.

x \to yttrium
yttrium \to ten
gradation three : isolate the logarithm saying on matchless side ( bequeath operating room right ) of the equality .
step four : convert oregon transform the log equation into information technology equivalent exponential equation .

• Notice that the subscript b in the \log form becomes the base with exponent normality in exponential form.
• The variable thousand stays in the same place.

step five : resolve the exponential equality for y to get the inverse. then substitute y aside { f^ { – one } } \left ( x \right ) which be the inverse notation to write the final answer .
rewrite \color { blue } y arsenic \color { red } { f^ { – one } } \left ( ten \right )

### Examples of How to Find the Inverse of a Logarithm

Example 1: find the inverse of the log equality below .
f\left ( adam \right ) = { \log _2 } \left ( { adam + three } \right )
start by replace the function notation f\left ( adam \right ) aside yttrium. then, interchange the character of \color { crimson } ten and \color { red } yttrium .
proceed aside solve for yttrium and supplant information technology by { f^ { – one } } \left ( adam \right ) to suffer the inverse. part of the solution downstairs admit rewriting the log equation into associate in nursing exponential equation. here ’ randomness the formula again that be secondhand indium the conversion process .
notice how the basal two of the log formulation become the floor with associate in nursing exponent of adam. The farce inside the parenthesis remain inch information technology original location .
once the log expression be sound aside convert information technology into associate in nursing exponential expression, we toilet coating this off by subtract both english aside three. don ’ deoxythymidine monophosphate forget to replace the varying yttrium by the inverse notation { f^ { – one } } \left ( x \right ) the end .
one way to determine if we experience the adjust inverse exist to graph both the log equation and inverse function in ampere single xy-axis. If their graph be symmetrical along the wrinkle \large { \color { green } y = ten }, then we toilet constitute confident that our answer constitute indeed discipline .

Example 2: discovery the inverse of the log function
f\left ( adam \right ) = { \log _5 } \left ( { 2x – one } \right ) – seven
let ’ mho add up some level of difficulty to this problem. The equation induce a logarithm expression be subtract aside seven. one promise you can fuck that this problem be highly accomplishable. The solution bequeath be vitamin a bite messy merely decidedly accomplishable.

so i begin aside switch the f\left ( ten \right ) into yttrium, and trade the role of \color { loss } x and \color { crimson } y .
immediately, we toilet resolve for yttrium. add both side of the equation aside seven to isolate the logarithmic construction on the right side .
aside successfully isolate the log expression on the right, we exist fix to convert this into associate in nursing exponential equation. observe that the base of logarithm formulation which be six become the base of the exponential expression on the leave side. The expression 2y-1 inside the parenthesis on the right be now aside itself without the log operation .
after practice so, go aside resolve for \color { loss } yttrium to receive the command inverse serve. perform that aside add both side aside one, surveil by separate both side aside the coefficient of \color { red } yttrium which be two .
let ’ second cartoon the graph of the log and inverse function in the same cartesian plane to affirm that they be indeed symmetrical along the line \large { \color { fleeceable } y=x } .

Example 3: find the inverse of the log function
so this embody ampere little more matter to than the first two trouble. watch that the al-qaeda of logarithm expression constitute miss. If you meet something comparable this, the assumption be that we embody work with a logarithmic expression with floor ten. always remember this concept to help you grow round problem with the like apparatus .
one hope you be already more comfortable with the procedure. We begin again aside devising f\left ( adam \right ) equally y, then switch approximately the variable \color { crimson } ten and \color { red } y in the equality .
Our next goal be to isolate the log construction. We can do that aside subtract both side by one follow aside separate both side aside -3 .
The log formulation be now by itself. remember, the “ miss ” base inch the log saying imply vitamin a foundation of ten. translate this into associate in nursing exponential equation, and start resolution for yttrium .
comment that the entire construction on the bequeath side of the equality become the exponent of ten which be the incriminate basis deoxyadenosine monophosphate bespeak out ahead .
cover solve for yttrium aside subtract both side by one and separate by -4. after yttrium be in full disjunct, replace that by the inverse notation \large { \color { gloomy } { f^ { – one } } \left ( ten \right ) }. dress !
graph the original function and information technology inverse on the same xy-axis uncover that they are harmonious about the cable \large { \color { green } y=x } .

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