MHB How Do You Solve for k in a Logarithmic Function with a Given Inverse?

  • Thread starter Thread starter karush
  • Start date Start date
  • Tags Tags
    Inverse Logarithm
AI Thread Summary
To solve for k in the logarithmic function f(x) = k log_2 x with the condition f^{-1}(1) = 8, the value of k is determined to be 1/3. This is derived by setting f(8) = 1, leading to the equation 3k = 1. Additionally, to find f^{-1}(2/3), the calculation shows that f^{-1}(2/3) equals 4, as derived from the logarithmic equation. The discussion highlights two approaches to arrive at the same conclusion regarding k and the inverse function. Overall, both methods confirm that k = 1/3 and f^{-1}(2/3) = 4.
karush
Gold Member
MHB
Messages
3,240
Reaction score
5
Let $$f(x) = k\ log_2 x$$

(a) Given that $$f^{-1}(1)=8$$, find the value of $$k$$

to get $$f^{-1}(x)$$ exchange $$x$$ and $$y$$

$$x=log_2 y^k$$

then convert to exponential form

$$2^x=y^k $$ then $$2^{\frac{x}{k}} = y$$

so for $$f^{-1}(1) = 2^{\frac{1}{k}}= 8=2^3$$ then $$\frac{1}{k}=3$$ so $$k=\frac{1}{3}$$

(b) find $$f^{-1}\bigg(\frac{2}{3}\bigg)=2^{\frac{2}{3}\frac{3}{1}}=2^2=4$$
 
Mathematics news on Phys.org
Re: inverse log and find k

a) Another approach would be to use that:

$$f^{-1}(1)=8\implies f(8)=1$$

and so:

$$f(8)=f\left(2^3 \right)=k\log_2\left(2^3 \right)=3k=1\,\therefore\,k=\frac{1}{3}$$

b) We could write:

$$f^{-1}\left(\frac{2}{3} \right)=x$$

$$f(x)=\frac{2}{3}$$

$$\frac{1}{3}\log_2(x)=\frac{2}{3}$$

$$\log_2(x)=2$$

$$x=2^2=4$$

Hence:

$$f^{-1}\left(\frac{2}{3} \right)=4$$
 
Re: inverse log and find k

well that was a better idea...:cool:
 
Re: inverse log and find k

karush said:
well that was a better idea...:cool:

I wouldn't say better, just different. :D
 
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Back
Top