Logarithms with the TI-Nspire

Calculator TI-Nspire keys used on this page

Key point 1 What a logarithm is

$$a = b^{x} \quad\Longleftrightarrow\quad \log_b a = x$$

• $\log x$ means $\log_{10} x$, and $\ln x$ means $\log_{\mathrm{e}} x$, where $\mathrm{e} \approx 2.718\,28$.
• $10^x$ is always positive, so you can only take the logarithm of a positive number.

Key point 2 Logs undo powers

Example 1: a log you can do in your head

Find the value of $\log 10\,000$ without a calculator.

• Let $x = \log 10\,000$, so $10^{x} = 10\,000$ (Key point 1).
• $10^{4} = 10\,000$, so $\log 10\,000 = 4$.
• Check on the TI-Nspire: ctrl + 10^x, then 10000, gives 4.

Practice: Exercise 1C Q1–8.

Example 2: an exact answer from a log equation

Find the exact value of $y$ if $\log(y-2) = 2$.

• Key point 1: $y - 2 = 10^{2} = 100$.
• $y = 102$.
• Check on the TI-Nspire: nSolve(log(y-2)=2, y) gives 102.

Practice: Exercise 1C Q9–11.

Example 3: rearranging with ln

Make $x$ the subject of $\ln(2x+5) = y$.

• Key point 1 with $b = \mathrm{e}$: $2x + 5 = \mathrm{e}^{y}$.
• $x = \dfrac{\mathrm{e}^{y} - 5}{2}$.
• This one has a letter in the answer, so it is algebra, not nSolve. To check, pick a value, say $y = 3$: the formula gives $x = 7.54$, and nSolve(ln(2x+5)=3, x) also gives 7.54.

Practice: Exercise 1C Q12–14.

Example 4: treat ln x like a letter

Simplify $\dfrac{5\ln x - 2\ln x}{\ln x}$.

• $5\ln x - 2\ln x = 3\ln x$, just as $5y - 2y = 3y$.
• $\dfrac{3\ln x}{\ln x} = 3$.

Practice: Exercise 1C Q15–17.

Example 5: undoing a log exactly

Find the exact value of $\mathrm{e}^{2\ln 3}$.

• $\mathrm{e}^{2\ln 3} = \left(\mathrm{e}^{\ln 3}\right)^{2}$ (law of exponents).
• $\mathrm{e}^{\ln 3} = 3$ (Key point 2), so the answer is $3^{2} = 9$.
• Check on the TI-Nspire: typing e^(2ln(3)) gives 9.

Practice: Exercise 1C Q18–25.

Example 6: evaluating a log on the calculator

Find $\ln 52\,900$, giving your answer in standard form to 3 significant figures.

• TI-Nspire: ctrl + e^x, then 52900, gives $10.876\,15\ldots$
• To 3 s.f. this is $10.9 = 1.09 \times 10^{1}$.

Practice: Exercise 1C Q26–28.

Example 7: solving an exponential equation with nSolve

Solve $\mathrm{e}^{x} = 7$, giving your answer to 3 significant figures.

• TI-Nspire: nSolve(e^x=7, x) gives $1.945\,91\ldots$
• $x = 1.95$ (3 s.f.).
• Why it works: Key point 1 says $x = \ln 7$, which is the number nSolve has found.

Example 8: nSolve first, exact form if asked

Solve $10^{x+1} = 45$.

• TI-Nspire: nSolve(10^(x+1)=45, x) gives $x = 0.653$ (3 s.f.).
• If the question asks for an exact answer, use Key point 1: $x + 1 = \log 45$, so $x = \log 45 - 1$.

Example 9: a harder equation is no harder on the calculator

Solve $\mathrm{e}^{2x-1} = 3\mathrm{e}^{x}$, giving your answer to 3 significant figures.

• TI-Nspire: nSolve(e^(2x-1)=3e^x, x) gives $x = 2.10$ (3 s.f.).
• The algebraic route needs several steps. With nSolve it is the same single command as Example 7.

Practice for Examples 7–9: Exercise 1C Q29–36, all with nSolve.

Finance Finance example (link to SL 1.4)

Question. Mia invests CHF 2000 in an account paying $4\%$ per year, compounded annually. Find the number of complete years until her investment first exceeds CHF 3000.

Method 1: nSolve - The value after $n$ years is $2000(1.04)^{n}$. - nSolve(2000*1.04^n=3000, n) gives $n = 10.34$. - Interest is added at the end of each year, so the answer is 11 complete years.

Method 2: Finance Solver - Enter $N = ?$, $I(\%) = 4$, $PV = -2000$, $Pmt = 0$, $FV = 3000$, $PpY = CpY = 1$. - Solving for $N$ gives $10.34$, so again 11 years.

Check: after 10 years the balance is CHF 2960.49, and after 11 years it is CHF 3078.91.

Question 1: acidity [6]

The pH of a solution is $\text{pH} = -\log[\mathrm{H^+}]$, where $[\mathrm{H^+}]$ is the hydrogen ion concentration in $\mathrm{mol\,dm^{-3}}$.

• (a) Lemon juice has $[\mathrm{H^+}] = 0.0063$. Find its pH. [2]
• (b) Human blood has a pH of $7.4$. Find $[\mathrm{H^+}]$ for blood. [2]
• (c) Normal rain has a pH of $5.6$, and a sample of acid rain has a pH of $4.3$. Find how many times more acidic the sample is. [2]

Question 2: earthquakes [7]

The magnitude of an earthquake is $M = \log A$, where $A$ is the amplitude in micrometres recorded 100 km from the epicentre.

• (a) Find $M$ when $A = 25\,000$. [2]
• (b) An earthquake has magnitude $6.2$. Find $A$. [2]
• (c) Find how many times larger the amplitude of a magnitude 7.1 earthquake is than that of a magnitude 6.2 earthquake. [2]
• (d) A news report says a 7.1 earthquake is "about 15% stronger" than a 6.2. Comment on this claim. [1]

Question 3: sound [8]

The sound level in decibels is $L = 10\log\left(\dfrac{I}{I_0}\right)$, where $I$ is the intensity in $\mathrm{W\,m^{-2}}$ and $I_0 = 10^{-12}\ \mathrm{W\,m^{-2}}$.

• (a) Normal conversation has $I = 10^{-6}$. Find $L$. [2]
• (b) A school canteen at lunchtime has $I = 3.2\times10^{-5}$. Find $L$. [2]
• (c) The risk of hearing damage begins at 85 dB. Find the corresponding intensity. [2]
• (d) A second, identical speaker is switched on, which doubles the intensity. Find the increase in $L$ and interpret your answer. [2]

Question 4: savings [3]

CHF 5000 is invested at 3.2% per year, compounded annually, so its value after $n$ years is $V = 5000(1.032)^{n}$.

• (a) Find the value of $n$ for which $V = 7000$. [2]
• (b) Hence state the number of complete years needed for the investment to reach CHF 7000. Justify your answer. [1]

Question 5: caffeine [6]

The amount of caffeine in a student's body $t$ hours after drinking a coffee is modelled by $C = 95\,\mathrm{e}^{-0.139t}$ mg.

• (a) Write down the amount of caffeine in the coffee. [1]
• (b) Find $C$ after 4 hours. [2]
• (c) Find the value of $t$ when $C = 20$. [2]
• (d) The student drinks the coffee at 15:00. Find the time at which $C$ falls below 20 mg, and comment on this in context. [1]