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# Exponents & Logs

Exponent & Log Laws, Solving Exponential & Logarithmic Equations…

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##### Question 1

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. $\log_6 9 + \log_6 4$. 

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

easy

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##### Question 2

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

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

easy

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##### Question 3

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. $\ln 3$ 

3. $\ln 48$ 

easy

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##### Question 4

no calculator

easy

[Maximum mark: 7]

Let $a=\ln 2$ and $b = \ln 10$. Write down the $\text{following}$ $\text{expressions}$ in terms of $a$ and $b$.

1. $\ln 20$ 

2. $\ln 5$ 

3. $\ln 160$ 

easy

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##### Question 5

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

3. $\log_a b$ 

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##### Question 6

no calculator

easy

[Maximum mark: 6]

Let $\log_3 p = u$, $\log_3 q = v$, $\log_3 r = w$. Write down the following expressions in terms of $u$, $v$ and $w$.

1. $\log_3\Big(\dfrac{r}{pq}\Big)$ 

2. $\log_3\Big(\dfrac{p^4r}{q^5}\Big)$ 

3. $\log_{pq} r$ 

easy

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##### Question 7

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. $\log_5 (25b)$ 

3. $\log_{25}b$ 

easy

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##### Question 8

no calculator

easy

[Maximum mark: 5]

Solve the equation $2\ln x=\ln 25 +6$, giving your answer in the form $x=ae^b$ where $a$, $b \in \mathbb{Z}^+$.

easy

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##### Question 9

no calculator

medium

[Maximum mark: 5]

Consider $b = \log_{80}81\times\log_{79}80\times\log_{78}79\times\dots\times\log_{3}4$. Given that $b\in\mathbb{Z}$, find the value of $b$.

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##### Question 10

no calculator

medium

[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 $a$.

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##### Question 11

no calculator

medium

[Maximum mark: 6]

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

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

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##### Question 12

no calculator

medium

[Maximum mark: 5]

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

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##### Question 13

no calculator

medium

[Maximum mark: 5]

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

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##### Question 14

calculator

medium

[Maximum mark: 6]

Given that $\log_a 2 = 5$.

1. Find the exact value of $\log_a 32$. 

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

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

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##### Question 15

calculator

medium

[Maximum mark: 6]

Given that $\log_b 3 = 10$.

1. Find the exact value of $\log_b 81$. 

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

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

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##### Question 16

no calculator

medium

[Maximum mark: 5]

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

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##### Question 17

no calculator

medium

[Maximum mark: 6]

1. Write down the value of

1. $\log_2 8$;

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

3. $\log_9 3$. 

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

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##### Question 18

no calculator

medium

[Maximum mark: 6]

1. Write down the value of

1. $\log_3 81$;

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

3. $\log_{25} 5$. 

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

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##### Question 19

no calculator

medium

[Maximum mark: 6]

1. Write the expression $3\ln 3 - \ln 9$ in the form $\ln a$, where $a \in \mathbb{Z}$. 

2. Hence, or otherwise, solve $3\ln 3 - \ln 9 = -\ln x$. 

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##### Question 20

no calculator

medium

[Maximum mark: 5]

Consider an arithmetic sequence with $u_{1}=5$ and $u_{6}=\log_3 32$.

Find the common difference of the sequence, expressing your answer in the form $\log_3 a$, where $a \in \mathbb{Q}$.

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##### Question 21

no calculator

medium

[Maximum mark: 5]

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

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##### Question 22

no calculator

medium

[Maximum mark: 5]

Solve $\log_4(x-12) + \log_4(x) = 3$, for $x > 12$.

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##### Question 23

no calculator

medium

[Maximum mark: 5]

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

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##### Question 24

no calculator

medium

[Maximum mark: 5]

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

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##### Question 25

no calculator

medium

[Maximum mark: 5]

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

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##### Question 26

no calculator

medium

[Maximum mark: 6]

Find the value of

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

2. $49^{\log_7 6}$. 

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##### Question 27

no calculator

medium

[Maximum mark: 6]

Find the value of

1. $\log_5 75 - \log_5 3$; 

2. $25^{\log_5 8}$. 

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##### Question 28

no calculator

medium

[Maximum mark: 8]

In an arithmetic sequence, $u_1 = \log_k (ab)$, $u_2 = \log_k(b)$, where $k > 1$ and $a,b > 0$.

1. Show that $d = -\log_k(a)$.

2. Let $a = k^4$ and $b = k^{16}$. Find the value of $\displaystyle \sum_{n=1}^{10} u_n$. 

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##### Question 29

no calculator

medium

[Maximum mark: 5]

Solve the equation $4\hspace{0.05em}\log_5 \sqrt{x} - \dfrac{1}{\log_3 5} = 3\log_5\left(3x^2\right)$, where $x>0$.

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##### Question 30

no calculator

hard

[Maximum mark: 5]

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

hard

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##### Question 31

no calculator

hard

[Maximum mark: 14]

The first two terms of an infinite geometric sequence, in order, are

$3\log_3x,\,\, 2\log_3x,\,\, \text{where x > 0.}$
1. Find the common ratio, $r$. 

2. Show that the sum of the infinite sequence is $9\log_3 x$. 

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

$\log_3x,\,\, \log_3 \dfrac{x}{3},\,\, \log_3\dfrac{x}{9},\,\, \text{where x > 0.}$
1. Find the common difference $d$, giving your answer as an integer. 

Let $S_6$ be the sum of the first $6$ terms of the arithmetic sequence.

1. Show that $S_6 = 6\log_3 x - 15$. 

2. Given that $S_6$ is equal to one third of the sum of the infinite geometric
sequence, find $x$, giving your answer in the form $a^p$ where $a,p \in \mathbb{Z}$. 

hard

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##### Question 32

no calculator

hard

[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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