Logarithmic scales - Answer key

$\log_{10} 1000 = 3$ , since $10^3 = 1000$ .
$\log_{10} 0.01 = -2$ , since $10^{-2} = 0.01$ .
$x = 10^4 = 10\,000$ .
$\dfrac{5\,000\,000}{500} = 10\,000 = 10^4$ . They are $4$ orders of magnitude apart.
$10^{30/10} = 10^3 = 1000$ times more intense.
Amplitude ratio: $10^{6.0 - 4.0} = 10^2 = 100$ times larger.
$\text{pH} = -\log_{10}(10^{-3}) = 3$ .
$10^{7-5} = 10^2 = 100$ times greater.

(a) Amplitude ratio: $10^{6.5 - 4.5} = 10^2 = 100$ times. (b) Energy ratio: $31.6^2 \approx 1000$ times.
Difference: $75 - 20 = 55$ dB. Intensity ratio: $10^{55/10} = 10^{5.5} \approx 316\,228$ times more intense.
The values span $4$ orders of magnitude ( $2$ to $20\,000$ ). On a linear scale, $2$ and $20$ would be indistinguishable near the axis while $20\,000$ dominates. A log scale spaces all five points evenly, revealing the constant factor-of- $10$ pattern.
Coffee is more acidic (lower pH). Concentration ratio: $10^{11.5 - 5} = 10^{6.5} \approx 3\,162\,278$ times more hydrogen ions in the coffee.
After $50$ years (5 doublings): $5000 \times 2^5 = 160\,000$ . On a log scale, exponential growth (constant doubling time) appears as a straight line because $\log(P) = \log(5000) + t \times \frac{\log 2}{10}$ , which is linear in $t$ .
Ratio $= 10^3 = 1000$ .

$\log_{10} 0$ is undefined because there is no power $n$ such that $10^n = 0$ (powers of $10$ are always positive). On a log-scale graph, zero cannot be plotted — the axis extends toward $-\infty$ in log-space as values approach zero. This means log scales can only represent strictly positive data.
Each magnitude step is a factor of $2.512$ . Over $5$ steps: $2.512^5 \approx 100$ . A magnitude- $1$ star is about $100$ times brighter than a magnitude- $6$ star.
The Richter scale is logarithmic, not linear. A magnitude $8$ quake has $10^{8-4} = 10^4 = 10\,000$ times the wave amplitude of a magnitude $4$ quake — not $2$ times. The student confused additive and multiplicative differences.
Example table (masses): electron $\approx 10^{-30}$ kg, grain of sand $\approx 10^{-6}$ kg, human $\approx 10^2$ kg, Earth $\approx 10^{24}$ kg, Sun $\approx 10^{30}$ kg. These span about $60$ orders of magnitude. A log scale is essential because a linear axis from $10^{-30}$ to $10^{30}$ would make all but the largest value invisible.

For $M = 5.0$ : $\log_{10} E = 1.5(5) + 4.8 = 12.3$ , so $E \approx 2.0 \times 10^{12}$ J. For $M = 8.0$ : $\log_{10} E = 1.5(8) + 4.8 = 16.8$ , so $E \approx 6.3 \times 10^{16}$ J. Ratio: $\dfrac{6.3 \times 10^{16}}{2.0 \times 10^{12}} \approx 31\,500 \approx 31.6^3$ (since $31.6^3 \approx 31\,623$ ). Confirmed.
Each source has intensity $I = I_0 \times 10^{80/10} = 10^8 I_0$ . Combined intensity $= 2 \times 10^8 I_0$ . Combined level $= 10\log_{10}(2 \times 10^8) = 10(\log_{10} 2 + 8) \approx 10(0.301 + 8) = 83.01$ dB. Doubling intensity adds about $3$ dB, not $80$ dB.
(a) $\log_{10}\!\left(\frac{10^8}{10^2}\right) = \log_{10}(10^6) = 6$ orders of magnitude. (b) A straight line, because $\log(\text{count})$ increases linearly with time for exponential growth. (c) Total growth factor $= 10^6$ over $24$ hours. Hourly factor $= (10^6)^{1/24} = 10^{0.25} \approx 1.778$ .
If $M_0$ increases by a factor of $1000 = 10^3$ , then $\log_{10}(M_0)$ increases by $3$ . So $M_w$ increases by $\frac{2}{3} \times 3 = 2$ units.

Tier 1