pH of a Soft Drink

A soft drink has a hydrogen ion concentration of $[H^+] = 3.2 \times 10^{-4}$ mol/L. Find its pH, rounded to two decimal places.

Write the formula: $\text{pH} = -\log[H^+] = -\log(0.00032)$ = $\text{pH} = -\log(0.00032)$

Evaluate the common logarithm: $\log(0.00032) \approx -3.49485$ = $\log(0.00032) \approx -3.49485$

Apply the minus sign: $\text{pH} = -(-3.49485) = 3.49485$ = $3.49485$

Round to two decimal places: The third decimal is $4$, so round down = $\text{pH} \approx 3.49$

Answer: $\text{pH} \approx 3.49$ (acidic, because it is below 7)

Decibel Level of a Vacuum Cleaner

A vacuum cleaner produces a sound $5\,000\,000$ times as intense as the threshold of hearing, so $\frac{I}{I_0} = 5\,000\,000$. Find its sound level in decibels, rounded to one decimal place.

Write the formula: $L = 10 \log\left(\frac{I}{I_0}\right) = 10 \log(5\,000\,000)$ = $L = 10 \log(5\,000\,000)$

Evaluate the logarithm: $5\,000\,000 = 5 \times 10^6$, so $\log(5\,000\,000) = \log 5 + 6 \approx 6.69897$ = $\approx 6.69897$

Multiply by 10: $L \approx 10 \times 6.69897 = 66.9897$ = $66.9897$

Round to one decimal place: The second decimal is $8$, so round up = $L \approx 67.0$ dB

Answer: $L \approx 67.0$ dB

Comparing Two Earthquakes

One earthquake measures $7.3$ on the Richter scale and another measures $5.1$. How many times larger is the amplitude of the stronger one? Round to the nearest whole number.

Find the difference in magnitude: $d = 7.3 - 5.1 = 2.2$ = $d = 2.2$

Turn the difference into a ratio: Each whole step on the scale multiplies the amplitude by $10$, so the ratio is $10^{d} = 10^{2.2}$ = $10^{2.2}$

Evaluate the power: $10^{2.2} = 10^2 \times 10^{0.2} \approx 100 \times 1.58489 = 158.489$ = $\approx 158.489$

Round to the nearest whole number: $158.489 \approx 158$ = $158$

Answer: The amplitude is about $158$ times larger (not $2.2$ times!)

How Long Until Savings Double?

You deposit 1000 dollars at $5\%$ interest compounded once a year. How long until the balance reaches 2000 dollars? Give $t$ rounded to two decimal places, then the number of full years needed.

Write the growth equation: $1000 \cdot 1.05^t = 2000$ = $1000 \cdot 1.05^t = 2000$

Isolate the power: Divide both sides by $1000$ = $1.05^t = 2$

Take ln of both sides and use the power rule: $\ln(1.05^t) = \ln 2 \Rightarrow t \cdot \ln 1.05 = \ln 2$ = $t = \frac{\ln 2}{\ln 1.05}$

Evaluate and round to two decimals: $t \approx \frac{0.693147}{0.048790} \approx 14.2067$ = $t \approx 14.21$ years

Interpret: interest is paid once a year: After 14 years: $1000 \cdot 1.05^{14} \approx 1979.93$ (not yet). After 15 years: $1000 \cdot 1.05^{15} \approx 2078.93$ = $15$ full years

Answer: $t \approx 14.21$ years, so the balance first reaches 2000 dollars after $15$ full years

Half-Life of a Medicine

A patient takes $400$ mg of a medicine with a half-life of $6$ hours. After how many hours will only $30$ mg remain? Round to one decimal place.

Write the half-life model: $400 \cdot \left(\frac{1}{2}\right)^{t/6} = 30$ = $400 \cdot 0.5^{t/6} = 30$

Isolate the power: $0.5^{t/6} = \frac{30}{400} = 0.075$ = $0.5^{t/6} = 0.075$

Take ln of both sides: $\frac{t}{6} \cdot \ln 0.5 = \ln 0.075$ = $\frac{t}{6} = \frac{\ln 0.075}{\ln 0.5} \approx 3.73697$

Multiply by 6 and round: $t \approx 6 \times 3.73697 \approx 22.4218$ = $t \approx 22.4$ hours

Answer: About $22.4$ hours (a bit less than 4 half-lives, since 4 half-lives would leave $25$ mg)

Mistake: Saying a magnitude 8 earthquake is "2 times" or "twice" as strong as a magnitude 6 earthquake

Why: The Richter scale is logarithmic: each whole step multiplies the amplitude by 10. Differences in magnitude are exponents, not multipliers.

Correct: Ratio of amplitudes $= 10^{8 - 6} = 10^2 = 100$ times.

Mistake: Forgetting the minus sign in $\text{pH} = -\log[H^+]$ and getting a negative pH like $-3.49$

Why: Concentrations such as $0.00032$ are less than 1, so their logarithm is negative. The minus sign in the formula makes the pH positive.

Correct: $\text{pH} = -\log(0.00032) = -(-3.49485) \approx 3.49$

Mistake: Using the rate instead of the growth factor: $t = \frac{\ln 2}{\ln 0.05}$

Why: The balance is multiplied by $1 + r = 1.05$ each year, not by $0.05$. Also $\ln 0.05$ is negative, which would give a negative time.

Correct: $t = \frac{\ln 2}{\ln(1.05)} \approx 14.21$ years. For decay use $1 - r$, for example $0.85$ for a $15\%$ loss.

Mistake: Rounding too early, for example writing $\ln 1.05 \approx 0.05$

Why: Small roundings inside a quotient grow: $\frac{0.6931}{0.05} \approx 13.86$ instead of $14.21$, which is wrong by about 4 months.

Correct: Keep full calculator precision (or at least 5 significant digits) and round only the final answer, as the problem says.

Acid Rain

Environmental scientists compare the acidity of rain with pH. Because pH is a logarithm, a small pH difference means a big difference in hydrogen ions.

Normal rain has a pH of about $5.6$. Rain with pH $4.6$ has $10^{1} = 10$ times as many hydrogen ions.

Hearing Protection at Concerts

Sound engineers and audiologists work in decibels. Ear plugs are rated by how much they cut the intensity of sound.

Dividing the intensity by $10$ lowers the level by $10$ dB, dividing by $100$ lowers it by $20$ dB.

Planning a Savings Goal

Financial planners use logarithms to find how many years an investment needs to reach a goal.

At $4\%$ per year, money grows by the factor $1.04$ each year, so the number of years is $t = \frac{\ln(\text{goal}/\text{start})}{\ln 1.04}$.

Q: Should I use log or ln to solve for time?

A: Either one works, as long as you use the same one on top and bottom: $\frac{\log 2}{\log 1.05} = \frac{\ln 2}{\ln 1.05} \approx 14.21$. Formulas for pH and decibels are defined with the common logarithm $\log$ (base 10), so use $\log$ there.

Q: Why is the answer 15 years if the calculator says 14.21?

A: The equation treats time as continuous, but a bank adds interest only once a year. After 14 years the balance is still below the target, so you need the 15th payment. Always ask whether the situation allows fractions of a year.

Q: Can a pH be negative or greater than 14?

A: Mathematically yes: $[H^+] = 2$ mol/L gives $\text{pH} = -\log 2 \approx -0.30$. In everyday chemistry, though, almost all solutions fall between 0 and 14.

Q: Is there a quick estimate for doubling time?

A: The "rule of 72" says doubling time is about $72 \div (\text{rate in }\%)$. At $5\%$: $72 \div 5 = 14.4$ years, close to the exact $14.21$. Use logarithms when the problem asks for an exact answer.