Question

Lori has determined a function f(x) that shows the exponential growth of the number of jeans Tim owns each year. Explain how the f−1^(x) can be found and what f−1^(250) means.

I need help , i want to be taught step by step. I even prepared my own function to use.
The function I chose to make for this problem is f(x)=5^x

Answer 1

@mathmate can you help walk me through this please?

Answer 2

the x is 250?? we know that f(x) is y so have u found why yet??

Answer 3

this is how far I've gone

f(x)=5^x
y=5^x
x=5^y
log5X=
log5 5^y

Answer 4

@domebotnos

Answer 5

@doc.brown

Answer 6

@mathmate

Answer 7

You were almost there!
y=5^x
exchange x and y
x=5^y
\(log_5 x = y\) by the definition of log
or
\(f^{-1}(x)=y=log_5x\)
so
\(f^{-1}(250)=y=log_5 250=log_5 5^3(2)=log_5 5^3+log_5 2=3+log_52\)
using properties of logarihms.

Answer 8

@mmathmate thanks a lot!~

Answer 9

@mathmate

Answer 10

You're welcome! :)

Answer 11

@mathmate wait, can you explain the middles step a little clearer for me?

Answer 12

\(f^{-1}(250)=y=log_5 250\).....from definition of inverse, substitute x=250
\(=log_5 5^3(2)\).................factorization of 250
\(=log_5 5^3+log_5 2\)..........log ab = log a + log b
\(=3log_5 5 + log_5 2\)
\(=3+log_52\)
...............from definition of logarithm \(log_a b=X \Leftrightarrow b=a^X\)
so \( log_5 5 \Leftrightarrow 5=5^1\) or \(log_5 5=1\)

Answer 13

@mathmate how did you get the 3 in the exponent?

Answer 14

in the factorization,
\(250 = 125*2 = 5*5*5 * 2 = 5^3 * 2\)
if that's the part you were querying.

Answer 15

@mathmate oh i get that part now, thanks. But how did you get to =3+log5 2?

Answer 16

The definition of log is
\(log_a b=X \Leftrightarrow b=a^X\)
That means the two expressions on each side of the double arrow are equivalent (for all values of a,b,X within the domain of log).
So if X=1, and a=2, then b=2^1=2, or \(log_2 2=1\)
Similarly,
if X=1, and a=5, then b=5^1=5, or \(log_5 5=1\)
In fact, in general
\(log_a a = 1\) for all values of a>0.
That is how \(log_5 5\) was evaluated to 1, which gives \(3log_5 5=3\)

Answer 17

@mathmate can you please simplify that for me? haha

Answer 18

Let's go line by line.
Are you familiar with the definition of log, shown on the second line of the last post?

Answer 19

The definition of log means that
if \(log_a b=X\) then it implies that \(b=a^X\)
AND
if \(b=a^X\) the it implies that \(log_a b=X\).

Answer 20

* for all all valid values of a, b and X.

Answer 21

@mathmate yes i understood this, whats after the equal sign is whats going to be exponent of the base correct? and the argument will be on the other side of the equal sign, right?

Answer 22

So
if 5^1 =5, what is \(log_5 5\) straight from the definition of log?

Answer 23

@mathmate can you rephrase that question?

Answer 24

Given 5^1=5 (which you already know), can you find what is
X=\(log_5 5\) using the definition of log:
\(log_a b=X \Leftrightarrow b=a^X\)

Answer 25

hint: put a=b=5

Answer 26

do you mean log5 5 =x to 5=5^x ?

Answer 27

Yes, what do you find for x?

Answer 28

is this where the factoring comes in and i choose the value of x to be 3?

Answer 29

You cannot \(choose\) x if 5^x=5, you need to \(solve\) for x because there is only one correct value of x.
Hints:
5^0=1
5^1=5
5^2=25
5^3=125

Answer 30

is it 125?

Answer 31

wait, its 2

Answer 32

5^125\(\ne\)5 !!!

Answer 33

\(5^2\ne5\) either.

Answer 34

its 3 because 5*5*5=125 correct?

Answer 35

The question we are trying to solve is
\(log_5 5=?\)
We are trying to solve an equivalent problem (using the definition of logarithm):
5^x=5
so 5*5*5=125 is not relevant.
\(5^x=5*....=5\) is relevant.

Answer 36

"Five to what power will give you five?"
is the equivalent problem we are trying to solve.

Answer 37

is it 1?

Answer 38

Exactly, that's how we got one in the last step.
Remember, Hints:
5^0=1
5^1=5
5^2=25
5^3=125
...

Answer 39

so just a check-up,
what is \(log_5 25\)?

Answer 40

17.47?

Answer 41

solve the equivalent problem for x
5^x=25

Answer 42

5^2

Answer 43

?

Answer 44

right*

Answer 45

so what is x?

Answer 46

x=2

Answer 47

?

Answer 48

right!
Let's put it together:
when you are given a problem to evaluate
\(\log_5 125=?\)
1. Assume \(\log_5 125=x\) and try to find x by the following.
2. formulate the equivalent problem using the definition of logarithm
\(log_5 125=x\) is the same problem as \(5^x=125\)
3. Solve for x in the last equation, using trial and error, or using the calculator. (x=3)
4. Go back to the original problem and state that \(log_5 125=3\).

Answer 49

Would that be a good summary to follow?
@cupcakes123

Answer 50

right!
Let's put it together:
when you are given a problem to evaluate
\(\log_5 125=?\)
1. Assume \(\log_5 125=x\) and try to find x by the following.
2. formulate the equivalent problem using the definition of logarithm
\(log_5 125=x\) is the same problem as \(5^x=125\)
3. Solve for x in the last equation, using trial and error, or using the calculator. (x=3)
4. Go back to the original problem and state that \(log_5 125=3\).

Answer 51

so it would be
log5^3+log5^2=250?

Answer 52

I am not sure where the expression is coming from, can you explain?

Answer 53

is it log5 5^3 +log5 2

Answer 54

Which problem are you solving? Can you please explain in detail?

Answer 55

so f^-1(x) would be equal to that?

Answer 56

im trying to say, does f^-1(250) is equal to log5 of 5^3?

Answer 57

+log5 of 2

Answer 58

\(f^{-1}(250)\) has already been solved above, and I extract the last line:
\(f^{-1}(250)=y=log_5 250=log_5 (5^3\times 2)=log_5 5^3+log_5 2=3+log_52\)
I have changed the parentheses to a multiplication symbol, hope that's more clear.

Answer 59

so, since they have the same base, which is 5, they cancel out and i multiply the arguments 5^3 times 2?

Answer 60

\(log_5 (125)=3\) because \(5^3=125\).

Answer 61

I don't understand what "they" stands for in "they" have the same base.

Answer 62

sorry lol i meant it has

Answer 63

In the last line, do you understand every step, which I broke down step by step above. Are you sure you understand every step, and why?

Answer 64

yes i understood that last line

Answer 65

250=5^3 \(\times\)2.
Since 2 is not an integer power of x, it has to remain as \(log_5 2\).
Since 125=5^3, we can simplify \(log_5 125=3\)
So
\(f^{-1}(250)=log_5 250=log_5 125 + log_5 2=3 + log_5 2\)

Answer 66

Remember log (ab) = log(a) + log(b), whatever the base of log.

Answer 67

Thank you very much for putting up with me. Im really sorry you have a lot of patience lol

thank you so mcuh for helping really appreciated. thank you

Answer 68

Are you sure you have understood?

Answer 69

yes, i understand now :)

Answer 70

Very well! Good night! P:)