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Logarithms Question

Hey guys, i’m trying to do this logarithm question here:
The value of a sculpture, £V, is modelled by the equation V = Ap^t, where A and p are constants and t is the number of years since the value of the painting was first recorded on 1st January 1960

The line I passes through the point (0,log10(20)) and (50,log10(2000))


b. Using your answer to part a or otherwise, find the values of A and p.


I’ve gotten to the point of equating the 2 equations like this:

1/25 t + log10 20 = t log10 p + log10 A

but i’m not sure how to eliminate the terms to find p and A
Original post by ramell
Hey guys, i’m trying to do this logarithm question here:
The value of a sculpture, £V, is modelled by the equation V = Ap^t, where A and p are constants and t is the number of years since the value of the painting was first recorded on 1st January 1960

The line I passes through the point (0,log10(20)) and (50,log10(2000))


b. Using your answer to part a or otherwise, find the values of A and p.


I’ve gotten to the point of equating the 2 equations like this:

1/25 t + log10 20 = t log10 p + log10 A

but i’m not sure how to eliminate the terms to find p and A


This is not right. I don't know why exactly you have equated two different quantities here.

From V=AptV=A p^t we take logs (with respect to base 10) to get logV=logA+tlogp\log V = \log A + t\log p.

Now use the data you have been provided. When t=0t=0 you have logV=log20\log V = \log 20 giving therefore

logV=logA+tlogp    log20=logA\log V = \log A + t\log p \implies \log 20 = \log A

Whereas when t=50t=50 you have logV=log2000\log V = \log 2000 therefore

logV=logA+tlogp    log2000=logA+50logp\log V = \log A + t\log p \implies \log 2000 = \log A + 50 \log p


So you have two relations {log20=logAlog2000=logA+50logp\begin{cases} \log 20 = \log A \\ \log 2000 = \log A + 50 \log p \end{cases}

Work out A,p from these.
(edited 7 months ago)

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