Exponentials and logarithms – Year 12 A level maths worksheet with answers to download as PDF

Solve each equation. Give non-exact answers to 3 significant figures.

1. \(2^x=32\) [1 mark]
2. \(3^{2x-1}=81\) [3 marks]
3. \(5^x=20\) [3 marks]
4. \(2^{3x-1}=7\) [4 marks]

Use the laws of logarithms.

1. Evaluate \(\log_240-\log_25\). [2 marks]
2. Evaluate \(2\log_36-\log_34\). [3 marks]
3. Evaluate \(\log_5\dfrac1{25}\) and \(\log_{16}4\). [3 marks]
4. Write \(3\log a+\tfrac12\log b-\log c\) as a single logarithm. [3 marks]

Solve each equation.

1. \(\log_3(x+2)=2\) [2 marks]
2. \(\log_2x+\log_2(x-2)=3\) [5 marks]
3. \(\log_{10}(3x+1)-\log_{10}(x-1)=1\) [4 marks]
4. \(\ln(2x-1)=3\), giving the answer to 3 significant figures [3 marks]

Solve each equation, giving exact answers in terms of natural logarithms where appropriate.

1. \(e^{2x}=12\) [2 marks]
2. \(e^{2x}-5e^x+6=0\) [4 marks]
3. \(3e^{-x}=7\) [3 marks]

Two experiments produced straight-line graphs after taking logarithms (base 10).

1. A graph of \(\log_{10}y\) against \(x\) has gradient 0.3 and intercept 0.7. Find \(a\) and \(b\) in \(y=ab^x\). [4 marks]
2. A graph of \(\log_{10}y\) against \(\log_{10}x\) has gradient 2.5 and intercept 0.6. Find \(a\) and \(n\) in \(y=ax^n\). [4 marks]

Consider the functions \(f(x)=2^x\) and \(g(x)=\log_2x\).

1. State the coordinates where each graph crosses an axis. [2 marks]
2. State the equation of the asymptote of each graph. [2 marks]
3. Explain why the graphs are reflections of each other in the line \(y=x\). [2 marks]
4. Solve \(g(x)=3\) and \(f(x)=3\). [3 marks]

Jamal invests £5 000 at 4% compound interest per year. The value after \(n\) years is \(V=5000\times1.04^n\).

1. Find the value after 10 years. [2 marks]
2. How many whole years until the investment has doubled? [4 marks]
3. What annual rate of interest would be needed to reach £8 000 in 10 years? [4 marks]
4. Explain why logarithms are needed for part (b). [1 mark]

A culture of bacteria grows according to \(N=200e^{0.3t}\), where \(t\) is the time in hours.

1. Find the number of bacteria at the start and after 8 hours (to the nearest whole number). [3 marks]
2. Find the doubling time in hours and minutes. [3 marks]
3. When will the culture reach 5 000 bacteria? [3 marks]
4. State one reason why the model cannot be accurate for large \(t\). [1 mark]

Carbon-14 decays so that the amount remaining after \(t\) years is \(N=N_0e^{-kt}\). Its half-life is 5 730 years.

1. Show that \(k=\dfrac{\ln2}{5730}\) and calculate \(k\) to 3 significant figures. [3 marks]
2. What percentage remains after 2 000 years? [2 marks]
3. A wooden bowl has 30% of its original carbon-14. Estimate its age. [3 marks]
4. Explain why a 40 000-year-old sample is hard to date. [1 mark]

The pH of a solution is \(\text{pH}=-\log_{10}[H^+]\) where \([H^+]\) is the hydrogen ion concentration in mol/litre.

1. Find the pH when \([H^+]=3.2\times10^{-5}\). [2 marks]
2. Find \([H^+]\) for blood with pH 7.4. [2 marks]
3. How many times more acidic (greater \([H^+]\)) is a solution of pH 3 than a solution of pH 5? [3 marks]
4. Why does a drop of 1 in pH correspond to multiplying \([H^+]\) by 10? [2 marks]

Sound level in decibels is \(L=10\log_{10}\left(\dfrac I{I_0}\right)\) where \(I\) is the intensity in W/m\(^2\) and \(I_0=10^{-12}\).

1. A road has intensity \(10^{-5}\) W/m\(^2\). Find its sound level. [2 marks]
2. By how many decibels does the level rise if the intensity doubles? [3 marks]
3. A level of 85 dB is a legal limit at work. Find the intensity. [3 marks]
4. Explain why ten identical machines are only 10 dB louder than one. [2 marks]

The number of followers of a new social media account (in hundreds) is recorded each week. The data are in the table, with the base-10 logarithm of each value.

1. Show that the points \((x,\log_{10}y)\) lie close to a straight line and find its gradient. [3 marks]
2. Hence find the constants \(a\) and \(b\) in the model \(y=ab^x\). [4 marks]
3. Predict the number of followers in week 10. [2 marks]
4. In which week does the model predict 100 000 followers (1 000 hundred)? [3 marks]

Biologists suggest that the daily energy use \(R\) (kcal) of a mammal of mass \(m\) kg follows \(R=am^n\). The table shows data for three animals.

1. Take logarithms of the relationship to show that \(\log R\) against \(\log m\) is a straight line. [2 marks]
2. Use the first two points to find \(n\) and \(a\). [4 marks]
3. Check that the third point fits. [2 marks]
4. Predict \(R\) for a 625 kg horse. [2 marks]

A new car costs £18 000. Its value after \(t\) years is modelled by \(V=18000\times0.82^t\).

1. What percentage of its value does the car lose each year? [1 mark]
2. Find the value after 3 years to the nearest £10. [2 marks]
3. After how many years will the car be worth less than £5 000? [4 marks]
4. How much does it lose in the first year? [1 mark]

The half-life of caffeine in the body is 5 hours. After drinking a 200 mg dose, the amount in the body is \(M=200\times\left(\tfrac12\right)^{t/5}\) mg, where \(t\) is in hours.

1. Write \(M\) in the form \(200e^{-kt}\), giving \(k\) to 3 significant figures. [3 marks]
2. How long until fewer than 10 mg remain? [4 marks]
3. At \(t=5\) a second 100 mg dose is taken. Find the total amount in the body at \(t=8\). [4 marks]
4. Explain why the total at \(t=8\) is almost the same from each dose. [1 mark]

Solve each problem.

1. \(\log_2x+\log_2y=5\) and \(x-y=4\), given \(x,y>0\). [6 marks]
2. Solve \(\log_2x=\log_x2\) (with \(x>0\), \(x\ne1\)). [4 marks]
3. Prove that \(\log_ab\times\log_ba=1\). [3 marks]

A town\'s population (in people) is recorded at two-year intervals. The model \(P=1200e^{kt}\) is proposed, with \(t\) in years.

1. Show that the values of \(\ln P\) increase by a constant amount per 2 years and find \(k\). [3 marks]
2. Find the time for the population to double. [3 marks]
3. Predict the population at \(t=10\) to the nearest 100. [2 marks]
4. The council plans for 5 000 people. When is this reached? [3 marks]

A patient takes a 150 mg tablet every morning. During each day the body keeps 70% of the drug. Immediately after the \(n\)th tablet the amount in the body is \(S_n=150+150(0.7)+\dots+150(0.7)^{n-1}\).

1. Check the amount after tablet 3 and identify the type of series. [2 marks]
2. Show that \(S_n=500(1-0.7^n)\). [2 marks]
3. What is the long-term maximum amount in the body? [2 marks]
4. After how many tablets is the amount within 1% of the maximum? [4 marks]

The graph shows \(y=\ln x\) (blue) and \(y=\ln(x-2)\) (red).

1. Describe the transformation from \(y=\ln x\) to \(y=\ln(x-2)\) and state the equation of the asymptote of each graph. [3 marks]
2. Find the exact \(x\)-coordinate where \(y=\ln(x-2)\) equals 2. [2 marks]
3. \(f(x)=3^x+1\). Find the inverse function \(f^{-1}(x)\) and its domain. [4 marks]
4. Solve \(\ln x-\ln(x-2)=\ln3\). [4 marks]

Solve each equation exactly.

1. \(3^{2x}-10\times3^x+9=0\) [4 marks]
2. \(\log_2(x-1)+\log_2(x+2)=2\) [4 marks]
3. \(2^x=3^{x-1}\), giving the answer to 3 significant figures [4 marks]