2.15 · Semi-log Plots

Try each one on paper first, then tap to check.

1. 1.

   Tell which graphs represent exponential functions and which do not. Then explain why.
   Graph 1 (normal axes).
   Show answer
2. 2.

   Tell which graphs represent exponential functions and which do not. Then explain why.
   Graph 2 (semi-log: $y$-axis 10, 100, 1000).
   Show answer
3. 3.

   Tell which graphs represent exponential functions and which do not. Then explain why.
   Graph 3 (semi-log: $y$-axis 10, 100, 1000).
   Show answer
4. 4a.

   Plot the following data on both graphs (normal and semi-log).

   $x$	2	4	5	8	9
   $y$	7	15	23	77	115

   Show answer
5. 4b.

   (4) Find a regression equation for the above data.
   Show answer
6. 4c.

   (4) Take the log of both sides and use log rules to create a linear function.
   Show answer
7. 4d.

   (4) Complete the table to find $\log y$.

   $x$	2	4	5	8	9
   $y$	7	15	23	77	115
   $\log y$

   Show answer
8. 4e.

   (4) Find a linear regression equation for $(x,\log y)$.
   Show answer
9. 5a.

   Plot the following data on both graphs (normal and semi-log).

   $x$	1	3	4	5	7
   $y$	65.0	32.5	16.3	8.2	1.0

   Show answer
10. 5b.

    (5) Find a regression equation for the above data.
    Show answer
11. 5c.

    (5) Take the log of both sides and use log rules to create a linear function.
    Show answer
12. 5d.

    (5) Complete the table to find $\log y$.

    $x$	1	3	4	5	7
    $y$	65.0	32.5	16.3	8.2	1.0
    $\log y$

    Show answer
13. 5e.

    (5) Find a linear regression equation for $(x,\log y)$.
    Show answer
14. 6a.

    Plot the following data on both graphs (normal and semi-log).

    $x$	1	3	4	5	7
    $y$	10.5	46.3	97.2	204.2	900.5

    Show answer
15. 6b.

    (6) Find a regression equation for the above data.
    Show answer
16. 6c.

    (6) Take the log of both sides and use log rules to create a linear function.
    Show answer
17. 6d.

    (6) Complete the table to find $\log y$.

    $x$	1	3	4	5	7
    $y$	10.5	46.3	97.2	204.2	900.5
    $\log y$

    Show answer
18. 6e.

    (6) Find a linear regression equation for $(x,\log y)$.
    Show answer
19. 7.

    The table gives values of $f$. Which graph could represent these data in a semi-log plot where the vertical axis is logarithmically scaled?
    (A) normal axes, points curving up to ~189
    (B) normal axes 0–5, points rising slowly from ~1.3 to ~2.3
    (C) $x$ normal (2, 4), $y$ log-scaled (1, 10, 100): points in a straight line
    (D) both axes log-scaled (1, 10 on $x$; 1, 10, 100 on $y$)

    $m$	1	2	3	4	5
    $f(m)$	18	32.4	58.3	104.9	188.9

    Show answer
20. 8A(i).

    (8) $g(x)=4\log x+\frac13\log x$, $h(x)=\frac{3^5\sqrt{3^x}}{3^2}$, $j(x)=\frac{(e^2)^x}{e^9}$, $k(x)=\log_2(x+4)+\log_2 x$.
    (A)(i) Rewrite $g(x)$ as a single logarithm without negative exponents.
    Show answer
21. 8A(ii).

    (8) (A)(ii) Rewrite $h(x)$ as a single base with an exponent.
    Show answer
22. 8B(i).

    (8) (B)(i) Solve $j(x)=e$ for values of $x$ in the domain of $j$.
    Show answer
23. 8B(ii).

    (8) (B)(ii) Solve $k(x)=5$ for values of $x$ in the domain of $k$.
    Show answer