A-Level Maths | Exponentials & Logs
Question 1
Express in the form $p\log_a x$
a) $\log_a x^5$
c) $\log_a x^7$
d) $\log_a x^{-4}$
e) $\log_a x^{1/2}$
b) $\frac{1}{2}\log_a x^{15}$
f) $3\log_a x^4$
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a) $5\log_a x$
c) $7\log_a x$
d) $-4\log_a x$
e) $\frac{1}{2}\log_a x$
b) $\frac{15}{2}\log_a x$
f) $12\log_a x$
Question 2
Simplify each of the following, giving the final answer as a single logarithm.
a) $\log_2 7+\log_2 2$
b) $\log_3 5+\log_3 2$
c) $\log_2 20-\log_2 4$
d) $\log_2 24-\log_2 8$
e) $\log_{10} 8+\log_{10} 5-\log_{10} 0.5$
f) $\log_6 2-(3\log_6 3+\log_6 0.25)$
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a) $\log_2 14$
b) $\log_3 10$
c) $\log_2 5$
d) $\log_2 3$
e) $\log_{10} 80$
f) $\log_6\left(\frac{8}{27}\right)$
Question 3
Simplify each of the following, giving the final answer as a single logarithm.
a) $3\log_5 2+\log_5 8$
b) $3\log_4 8-3\log_4 6$
c) $2\log_6 8-5\log_6 2$
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a) $\log_5 64$
b) $\log_4\left(\frac{64}{27}\right)$
c) $\log_6 2$
Question 4
Solve each of the following exponential equations.
a) $3^x=11$
b) $4^x=200$
c) $2^x=50$
d) $5^x=30$
e) $4^{y-1}=18$
f) $2^{2z+1}=80$
g) $5^{3w}=30$
h) $12^{3t+4}=75$
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a) $x \approx 2.18$
b) $x \approx 3.82$
c) $x \approx 5.64$
d) $x \approx 2.11$
e) $y \approx 3.08$
f) $z \approx 2.66$
g) $w \approx 0.704$
h) $t \approx -0.754$
Question 5
Solve each equation, giving your answers correct to 3 significant figures.
a) $\log_3 x=1.8$
b) $\log_5 x=-0.3$
c) $\log_2 x=3.5$
d) $\log_7 x=-1.2$
e) $\log_4 x=2.6$
f) $\log_6 x=0.9$
g) $\ln x=2.1$
h) $\ln x=-0.4$
i) $\ln x=0.7$
j) $\ln x=3.2$
k) $\ln x=-1.5$
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a) $x \approx 7.22$
b) $x \approx 0.617$
c) $x \approx 11.3$
d) $x \approx 0.0968$
e) $x \approx 36.8$
f) $x \approx 5.02$
g) $x \approx 8.17$
h) $x \approx 0.670$
i) $x \approx 2.01$
j) $x \approx 24.5$
k) $x \approx 0.223$
Question 6
a) $\log_2(x+1)-\log_2 x=\log_2 3$
b) $\log_a x=\log_a 3+\log_a(2x-1)$
c) $\log_a(2x+7)=\log_a x+2\log_a 3$
d) $\log_a(3x+10)-\log_a x=2\log_a 3$
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a) $x=0.5$
b) $x=0.6$
c) $x=1$
d) $x=\frac{5}{3}$
Question 7
a) $\log_3(3x+4)-\log_3 x=2$
b) $\log_5(4x+3)-\log_5(x-1)=2$
c) $\log_5(4x+7)-\log_5 x=2$
d) $\log_2(2x+1)=2+\log_2 x$
e) $\log_3(4x+1)-\log_3(x-1)=2$
f) $\log_2(3x+4)-\log_2 x=3$
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a) $x=\frac{2}{3}$
b) $x=\frac{4}{3}$
c) $x=\frac{1}{3}$
d) $x=\frac{1}{2}$
e) $x=2$
f) $x=\frac{4}{5}$
Question 8
The value of a rare painting, $V$ pounds, is modelled by $V = pq^t$, where $p$ and $q$ are constants and $t$ is the number of years since the value was first recorded on 1st January 1980.
a) Write $V = pq^t$ in the form $y = mx + c$.
b) The gradient of the line is $0.05$ and the $y$-intercept is $4.8$. Find the value of $p$ and the value of $q$, each to 4 significant figures.
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a) $\log_{10}V = t\log_{10}q + \log_{10}p$
b) $\log_{10}p = 4.8 \implies p = 63100$ (4 s.f.). $\log_{10}q = 0.05 \implies q = 1.122$ (4 s.f.).
Question 9
The total number of views, $V$, of an online advert in the first $t$ days is modelled by $V = ab^t$, where $a$ and $b$ are constants.
a) Write $V = ab^t$ in the form $y = mx + c$.
b) The gradient of the line is $0.072$ and the $y$-intercept is $2.379$. Find $a$ to the nearest whole number and $b$ to 3 significant figures.
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a) $\log_{10}V = t\log_{10}b + \log_{10}a$
b) $\log_{10}a = 2.379 \implies a = 239$ (nearest whole number). $\log_{10}b = 0.072 \implies b = 1.18$ (3 s.f.).
Question 10
The resting heart rate, $h$ beats per minute, of a mammal is modelled by $h = pm^q$, where $p$ and $q$ are constants and $m$ is the mass of the mammal in kg.
a) Write $h = pm^q$ in the form $y = mx + c$.
b) The gradient of the line is $-0.235$ and the $y$-intercept is $2.25$. Find the value of $p$ and the value of $q$, each to 3 significant figures.
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a) $\log_{10}h = q\log_{10}m + \log_{10}p$
b) $q = -0.235$ (3 s.f.). $\log_{10}p = 2.25 \implies p = 178$ (3 s.f.).
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