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Thread: f(x)=b*a^x

  1. #1
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    f(x)=b*a^x

    Here is a question I want you to look into. I want to see where I have gone wrong.

    In 1948 there was 823 cancerpatients. I 1993 there was 5707 cancerpatients. Find the values a and b. Facit: a = 1.045 b = 770

    This is what I have done to find these values:

    1993-1948= 45 years

    823=b*a^0 --> 823/(a^0) = b

    5707=b*a^45 --> 5707/(a^45) = b

    823/(a^0) = 5707/(a^45) --> (a^45)/(a^0) = 5707/823

    a^0 = 1

    a = 45sq.root(5707/823) = 1.044

    Now I have the value for a. Now for b:

    5707/(1.044^45) = 822
    eller
    823/(1.044^0) = 823

    How come b is not the same as the facit? What did I do wrong?
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  2. #2
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    Hello, No Logic Sense!

    I see absolutely nothing wrong with your work . . .


    In 1948 there was 823 cancer patients.
    In 1993 there was 5707 cancer patients.
    Find the values $\displaystyle a\text{ and }b.$
    . . Facit: $\displaystyle a = 1.045\;{\color{red}?}\;\; b = 770\;\;{\color{red}??}$
    We have: .$\displaystyle f(t) \:=\:ba^t$

    And we have: . $\displaystyle \begin{array}{cc}1948\:(t = 0) & 823 \\ 1993\:(t = 45) & 5707 \end{array}$

    $\displaystyle f(0) = 823\quad\Rightarrow\quad ba^0 \:=\:823 \quad\Rightarrow\quad\boxed{ b \:=\:823}$

    The function (so far) is: .$\displaystyle f(x) \:=\:823a^t$

    $\displaystyle f(45) = 5707\quad\Rightarrow\quad 823a^{45} \:=\:5707\quad\Rightarrow\quad a^{45} \:=\:\frac{5707}{823} $

    Then: .$\displaystyle a \:=\:\left(\frac{5707}{823}\right)^{\frac{1}{45}} \:=\:1.043972521 \quad\Rightarrow\quad\boxed{ a \:\approx\:1.044}$


    Therefore, the function is: .$\displaystyle \boxed{f(t) \;=\;823(1.044)^t} $

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  3. #3
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    My teacher said that it has something to do with drawing the function on a logarithm paper. Apparently if you do that, you will get the b value to 770.
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  4. #4
    GAMMA Mathematics
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    Quote Originally Posted by No Logic Sense View Post
    My teacher said that it has something to do with drawing the function on a logarithm paper. Apparently if you do that, you will get the b value to 770.
    logarithm paper?
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  5. #5
    Lord of certain Rings
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    Quote Originally Posted by colby2152 View Post
    logarithm paper?
    I think he means the log graph sheet, where the x axis is in logarithmic scale and y axis is normal.
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  6. #6
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    Most "log graph paper" has a normal x-axis and a logarithmic y-axis.
    . . Then an exponential function becomes a line.


    We have .$\displaystyle y \;=\;823(1.044)^x$

    Take logs: .$\displaystyle \ln(y) \;=\;\ln\bigg[823(1.044)x\bigg] \;=\;\ln(823) + \ln(1.044)^x \;=\;\ln(823) + x\!\cdot\!\ln(1.044)$

    Then we have: .$\displaystyle \ln(y) \;=\;[\ln(1.044)]\!\cdot\! x + \ln(823)$

    Let $\displaystyle m = \ln(1.044)$ ... a constant.
    Let $\displaystyle b = \ln(823)$ ... another constant.

    The equation becomes: . $\displaystyle \ln(y) \;=\; mx + b \quad\hdots\quad See?$

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