Subjects

# Exponents & Logs

Exponent & Log Laws, Solving Exponential & Logarithmic Equations…

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##### Question 1

no calculator

easy

[Maximum mark: 7]

Find the value of each of the following, giving your answer as an integer.

1. $\log_{10} 100$. [2]

2. $\log_{10} 50 + \log_{10} 2$. [2]

3. $\log_{10} 4 - \log_{10} 40$. [3]

easy

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##### Question 2

no calculator

easy

[Maximum mark: 6]

Find the value of each of the following, giving your answer as an integer.

1. $\log_6 6$. [2]

2. $\log_6 9 + \log_6 4$. [2]

3. $\log_6 72 - \log_6 2$. [2]

easy

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##### Question 3

no calculator

easy

[Maximum mark: 6]

Let $\log_2 a = p$, $\log_2 b = q$, $\log_2 c = r$. Write down the following expressions in terms of $p$, $q$ and $r$.

1. $\log_2\Big(\dfrac{ab}{c}\Big)$ [2]

2. $\log_2\Big(\dfrac{a^2c}{b^3}\Big)$ [2]

3. $\log_a b$ [2]

easy

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##### Question 4

no calculator

easy

[Maximum mark: 5]

Consider $a = \log_{63}64\times\log_{62}63\times\log_{61}62\times\dots\times\log_{2}3$. Given that $a\in\mathbb{Z}$, find the value of $a$.

easy

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##### Question 5

no calculator

easy

[Maximum mark: 7]

Let $p=\ln 2$ and $q = \ln 6$. Write down the following expressions in terms of $p$ and $q$.

1. $\ln 12$ [2]

2. $\ln 3$ [2]

3. $\ln 48$ [3]

easy

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##### Question 6

no calculator

easy

[Maximum mark: 5]

Solve the equation $\log_5 x - \log_5 4 = 2 + \log_5 3$ for $x$.

easy

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##### Question 7

calculator

easy

[Maximum mark: 6]

Given that $\log_a 2 = 5$.

1. Find the exact value of $\log_a 32$. [2]

2. Find the exact value of $\log_{\sqrt{a}} 2$. [2]

3. Find the value of $a$, giving your answer correct to $3$ significant figures. [2]

easy

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##### Question 8

no calculator

easy

[Maximum mark: 5]

Solve the equation $\log_3 x - \log_3 5 = 1 + \log_3 4$ for $x$.

easy

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##### Question 9

no calculator

easy

[Maximum mark: 6]

Let $a = \log_5b$, where $b > 0$. Write down each of the following expressions
in terms of $a$.

1. $\log_5b^4$ [2]

2. $\log_5 (25b)$ [2]

3. $\log_{25}b$ [2]

easy

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##### Question 10

calculator

easy

[Maximum mark: 6]

Given that $\log_b 3 = 10$.

1. Find the exact value of $\log_b 81$. [2]

2. Find the exact value of $\log_{b^2} 3$. [2]

3. Find the value of $b$, giving your answer correct to $3$ significant figures. [2]

easy

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##### Question 11

no calculator

easy

[Maximum mark: 6]

1. Write the expression $4\ln 2 - \ln 8$ in the form of $\ln k$, where $k \in \mathbb{Z}$. [3]

2. Hence, or otherwise, solve $4\ln 2 - \ln 8 = -\ln (2x)$. [3]

easy

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##### Question 12

no calculator

easy

[Maximum mark: 5]

Solve the equation $\log_3(x^2-4x+4) = 1 + \log_3(x-2)$.

easy

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##### Question 13

no calculator

easy

[Maximum mark: 5]

Find the values of $x$ when $27^{x+2} = \left(\dfrac{1}{9}\right)^{2x+4}$.

easy

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##### Question 14

no calculator

easy

[Maximum mark: 5]

Solve the equation $\log_2(x^2-2x+1) = 1 + \log_2(x-1)$.

easy

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##### Question 15

no calculator

easy

[Maximum mark: 6]

1. Write down the value of

1. $\log_2 8$;

2. $\log_5\Big(\dfrac{1}{25}\Big)$;

3. $\log_9 3$. [3]

2. Hence solve $\log_2 8 + \log_5\Big(\dfrac{1}{25}\Big) + \log_9 3 = \log_{16} x$.[3]

easy

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##### Question 16

no calculator

easy

[Maximum mark: 6]

1. Write down the value of

1. $\log_3 81$;

2. $\log_2\Big(\dfrac{1}{8}\Big)$;

3. $\log_{25} 5$. [3]

2. Hence solve $\log_3 81 + \log_2\Big(\dfrac{1}{8}\Big) + \log_{25} 5 = \log_{9} x$.[3]

easy

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##### Question 17

no calculator

easy

[Maximum mark: 5]

Solve $\log_6(x) + \log_6(x-5) = 2$, for $x > 5$.

easy

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##### Question 18

no calculator

easy

[Maximum mark: 5]

Find the values of $x$ when $25^{x^2-2x} = \left(\dfrac{1}{125}\right)^{4x+2}$.

easy

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##### Question 19

no calculator

easy

[Maximum mark: 5]

Solve the equation $9^x + 2\cdot3^{x+1} = 1$.

easy

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##### Question 20

no calculator

easy

[Maximum mark: 6]

Find the value of

1. $\log_5 75 - \log_5 3$; [2]

2. $25^{\log_5 8}$. [4]

easy

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##### Question 21

no calculator

easy

[Maximum mark: 6]

Find the value of

1. $\log_7 98 - \log_7 2$; [2]

2. $49^{\log_7 6}$. [4]

easy

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##### Question 22

no calculator

medium

[Maximum mark: 5]

Solve the equation $15^{4a} = 81^{a+2}$ for $a$. Express your answer in terms of $\ln 3$ and $\ln 5$.

medium

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##### Question 23

no calculator

medium

[Maximum mark: 7]

Consider $f(x) = \log_k(8x-2x^2)$, for $0 < x < 4$, where $k > 0$.

The equation $f(x) = 3$ has exactly one solution. Find the value of $k$.

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##### Question 24

no calculator

medium

[Maximum mark: 6]

Solve $\log_{\sqrt{3}}(\sin x) - \log_{\sqrt{3}}(\cos x) = 1$, for $0 < x < \dfrac{\pi}{2}$.

medium

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##### Question 25

calculator

medium

[Maximum mark: 5]

Solve the equation $14^{6x} = 64^{x+3}$ for $x$. Express your answer in terms of $\ln 2$ and $\ln 7$.

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##### Question 26

no calculator

medium

[Maximum mark: 15]

The equation $e^{2x} - 12e^x = -32$ has two solutions, $x_1$ and $x_2$.

1. Find the value of $x_1$ and the value of $x_2$.[5]

A second equation, $2\left(\log_{\,9}x\right)\left(\log_{\,3}x\right) - 6\log_{\,9}x - 2\log_{\,3}x = -6$, also has two solutions, $x_3$ and $x_4$.

1. Show that this second equation can be expressed as

\begin{align*} \left(\log_{\,3}x\right)^2 -5\log_{\,3}x + 6 = 0 \end{align*}
2. Hence find the value of $x_3$ and the value of $x_4$. [7]

1. Given that $x_1 + x_2 = a(x_3 + x_4)$, find the value of $a$. Give your answer in the form $b\ln c$, where $b,c \in \mathbb{R}$.[3]

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##### Question 27

no calculator

medium

[Maximum mark: 5]

Find the integer values of $a$ and $b$ for which

\begin{aligned} a + b\log_4 7 + 60\log_{8} 14 &= 0.\end{aligned}

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##### Question 28

no calculator

medium

[Maximum mark: 18]

The first three terms of an infinite sequence, in order, are

$2\ln x,\,\, q\ln x,\,\, \ln \sqrt{x}\,\,\, \text{ where \ x > 0.}$

First consider the case in which the series is geometric.

1. Find the possible values of $q$.

2. Hence or otherwise, show that the series is convergent. [3]

1. Given that $q>0$ and $S_\infty=8\ln{3}$, find the value of $x$. [3]

Now suppose that the series is arithmetic.

1. Show that $q=\dfrac{5}{4}$.

2. Write down the common difference in the form $m\ln x$, where $m \in \mathbb{Q}$. [4]

1. Given that the sum of the first $n$ terms of the sequence is $\ln \sqrt{x^5}$, find the value of $n$. [8]

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##### Question 29

no calculator

medium

[Maximum mark: 8]

1. Show that $\log_{16}(\cos 2x + 7) = \log_4 \sqrt{\cos 2x + 7}$. [3]

2. Hence, or otherwise, solve $\log_4(\sqrt{10}\cos x) = \log_{16}(\cos 2x + 7)$, for $0 < x < \dfrac{\pi}{2}$.[5]

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##### Question 30

no calculator

medium

[Maximum mark: 7]

Solve the simultaneous equations:

\begin{aligned} 1 + 2\log_5 x &= \log_5 7y \\[6pt] \log_7 (6x-1) &= 1 + \log_7 y\end{aligned}

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