Exponentials & Logarithms · Guided Practice

Exponentials & Logarithms

23 questions with answers and video solutions. Try each one before revealing the answer.

1
Evaluate log⁡464\displaystyle \log_4 64.
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No hints on this one — its full written solution is with the answer.

2
Evaluate log⁡40+2log⁡5\displaystyle \log 40+2\log 5.
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No hints on this one — its full written solution is with the answer.

3
Evaluate log⁡912−2log⁡92\displaystyle \log_9 12-2\log_9 2.
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No hints on this one — its full written solution is with the answer.

4
Evaluate 2log⁡63+13log⁡664\displaystyle 2\log_6 3+\frac{1}{3}\log_6 64.
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No hints on this one — its full written solution is with the answer.

5
Solve log⁡6x+log⁡6(x+5)=2\displaystyle \log_6 x+\log_6 (x+5)=2, where x>0\displaystyle x > 0.
Video solution coming soon

No hints on this one — its full written solution is with the answer.

6
Given that log⁡a45−log⁡a5=12\displaystyle \log_a 45-\log_a 5=\frac{1}{2}, find the value of a\displaystyle a.
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No hints on this one — its full written solution is with the answer.

7
Solve 62x+1=19\displaystyle 6^{2x+1}=19, correct to 3 significant figures.
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No hints on this one — its full written solution is with the answer.

8
The number of people infected by a virus is described by the exponential formula I(t)=5e0.07t\displaystyle I(t)=5e^{0.07t}, where t\displaystyle t is the time in days.
(a) How many people were infected on day zero?
(b) To the nearest day, how long will it take for the number of infections to double?
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No hints on this one — its full written solution is with the answer.

9
Two variables x\displaystyle x and y\displaystyle y are connected by the equation y=abx\displaystyle y=ab^{x}. The graph of log⁡5y\displaystyle \log_5 y against x\displaystyle x is a straight line passing through (0,−1)\displaystyle (0,-1) and (4,1)\displaystyle (4,1).
Find the values of a\displaystyle a and b\displaystyle b.
Video solution coming soon

No hints on this one — its full written solution is with the answer.

10
Two variables x\displaystyle x and y\displaystyle y are connected by the equation y=kxn\displaystyle y=kx^{n}. The graph of log⁡3y\displaystyle \log_3 y against log⁡3x\displaystyle \log_3 x is a straight line passing through (−4,0)\displaystyle (-4,0) and (0,3)\displaystyle (0,3).
Find the values of k\displaystyle k and n\displaystyle n.
Video solution coming soon

No hints on this one — its full written solution is with the answer.

11
2017 P1 Q12
3 Marks
Given that
log⁡a36−log⁡a4=12\displaystyle \log_{a}36-\log_{a}4=\frac{1}{2}
find the value of a\displaystyle a.
12
2018 P1 Q6
3 Marks

Find the value of
log⁡5250−13log⁡58\displaystyle \log_{5}250-\frac{1}{3}\log_{5}8

13
2021 P1 Q16
Evaluate log⁡26+log⁡212−2log⁡23\displaystyle \log_2 6+\log_2 12-2\log_2 3.
Video solution coming soon

No hints on this one — its full written solution is with the answer.

14
2022 P1 Q2
3 Marks
Evaluate
2log⁡36−log⁡34\displaystyle 2 \log_{3} 6-\log_{3} 4
15
2023 P1 Q3
3 Marks
Solve
log⁡5x−log⁡53=2\displaystyle \log_{5}x-\log_{5}3=2
16
2024 P1 Q9
3 Marks

Express
log⁡a5+log⁡a80−2log⁡a10\displaystyle \log_{a}5+\log_{a}80-2\log_{a}10
in the form log⁡ak\displaystyle \log_{a}k where k is a positive integer.

17
2024 P2 Q11
(1, 4)5 Marks

The number of electric vehicles worldwide can be modelled by
N=6.8ekt\displaystyle N=6.8e^{kt}
where:
•  N is the estimated number of vehicles in millions
•  t is the number of years since the end of 2020
•  k is a constant.
(a)  Use the model to estimate the number of electric vehicles worldwide at the end of 2020.
At the end of 2030, it is estimated there will be 125 million electric vehicles worldwide.
(b)  Determine the value of k.

18
2025 P1 Q4
3 Marks
Evaluate
3log⁡32+log⁡3124.\displaystyle 3\log_{3}2+\log_{3}\frac{1}{24}.
19
2025 P1 Q8
3 Marks
Given that
log⁡a75=2+log⁡a3\displaystyle \log_{a}75=2+\log_{a}3 where a>0\displaystyle a \gt 0,
find the value of a.\displaystyle a.
20
2025 P2 Q13
(1, 4)5 Marks

A radioactive substance, which has been collected, decays over time.
The mass of the radioactive substance remaining is modelled by
M=150e−0.0054t\displaystyle M=150e^{-0.0054t}
where M is the mass, in micrograms, t years after the radioactive substance was collected.
(a)  Determine the initial mass of the radioactive substance.
(b)  Calculate the time taken for the mass of the radioactive substance to decay to 120 micrograms.

21
2026 P1 Q8
4 Marks
Solve log⁡2x+log⁡2(x+2)=3\displaystyle \log_{2}x+\log_{2}(x+2)=3, where x>0.\displaystyle x \gt 0.
22
2026 P2 Q11
5 Marks

Two variables x and y are connected by the equation y=abx.\displaystyle y=ab^{x}.
The graph of log⁡4y\displaystyle \log_{4}y against x is a straight line as shown.

Straight line graph of log base 4 of y against x

Find the values of a and b.

23
2026 P2 Q14
(1, 4)5 Marks

During a metal-working process a heated steel rod is cooled by dropping it into a container of oil.
The temperature of the rod is given by
T=800e−0.124t+25,\displaystyle T=800e^{-0.124t}+25,
where T is the temperature of the rod, in degrees Celsius, t seconds after it is dropped into the oil.
(a)  Determine the temperature of the rod as it is dropped into the oil.
(b)  Calculate the time taken for the temperature of the rod to cool to 150∘\displaystyle 150^{\circ}C.

Practice questions courtesy of Maths.scot. Full written solutions are on his site — the links above go straight to them.

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