Semi-log Plots

On a logarithmic vertical scale, equal vertical steps are equal factors. A straight line rises by equal steps over equal times, so the count is multiplied by the same factor each year: exponential growth.

• Correct: It grows exponentially, by the same factor in equal times: Straight on a semi-log plot means $\log y$ is linear in $t$, so $y$ is exponential.
• It grows linearly: equal time steps add the same number of users: Linear data would bend (concave down) on a log scale, because equal additions are smaller and smaller factors.
• It grows like a quadratic: its second differences are constant: A quadratic's logarithm is not linear in $t$, so it would not make a straight line.
• It grows logarithmically: it rises by equal steps as $t$ doubles: Logarithmic growth would bend even more on a log scale.

$B$ is one labeled step (one power of $10$) above $A$: from $20$ to $200$, a factor of $10$, over $4$ years.

• Correct: It is multiplied by $10$: $\frac{200}{20} = 10$.
• It increases by $10$ users: The count goes from $20$ to $200$, up $180$; on a log scale, equal steps are factors, not differences.
• It is multiplied by $4$: $4$ is the number of years, not the factor.
• It is multiplied by $100$: A factor of $100$ is two labeled steps, from $A$ to $C$; $B$ is one step above $A$.

$\frac{\log 2{,}000 - \log 20}{8 - 0}$$= \frac{3.301 - 1.301}{8}$$= \frac{2}{8}$$= 0.25$ per year.

• Correct: $0.25$: $\log y$ rises $2$ (two powers of $10$) in $8$ years.
• $247.5$: That is $\frac{2{,}000 - 20}{8}$, the slope on an ordinary scale; this axis shows $\log y$.
• $10$: $10$ is the factor over $4$ years, not the change in $\log y$ per year.
• $1.778$: That is $10^{0.25}$, the growth factor $b$; the slope is its logarithm, $0.25$.

$\log y$ increases by $0.3$ for each step of $1$ in $t$, so $\log y$ is linear in $t$ and $y$ is exponential.

• Correct: $y$ is exponential in $t$: $\log y$ linear in $t$ means $y = a \cdot b^t$.
• $y$ is linear in $t$: It is $\log y$, not $y$, that grows by equal amounts.
• $y$ is logarithmic in $t$: A logarithmic $y$ would make $y$ itself change by equal steps over equal factors of $t$.
• $y$ is quadratic in $t$: $\log y$ has constant first differences, so it is linear, and $y$ is exponential.

The starting value at $t = 0$ is $0.5$ and the constant change is $0.3$ per unit of $t$: $\log y = 0.5 + 0.3t$.

• Answer: $\log y =$ $0.5 + 0.3t$

$\log y = 0.5 + 0.3t$, so $y$$= 10^{0.5 + 0.3t}$$= 10^{0.5} \cdot 10^{0.3t}$$= 10^{0.5} \cdot (10^{0.3})^t$.

• Correct: $y = 10^{0.5 + 0.3t}$: Undo the logarithm with base $10$, then split the exponent.
• $y = 0.5 + 0.3t$: That is $\log y$, not $y$.
• $y = 0.5 \cdot (0.3)^t$: The intercept and slope are logarithms of $a$ and $b$, so $a = 10^{0.5}$ and $b = 10^{0.3}$.
• $y = 10^{0.5} + 10^{0.3t}$: $10^{0.5 + 0.3t}$ is a product of powers, not a sum.