Solve exponential equations

A question is this type if and only if it requires solving an equation where the variable appears in an exponent, such as 3^(6x-3) = 81 or 4^(2x+1) = 8^(4x), typically by expressing both sides with the same base.

16 questions · Easy -1.0

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Edexcel C12 2016 June Q3
7 marks Moderate -0.8
3. Answer this question without a calculator, showing all your working and giving your answers in their simplest form.
  1. Solve the equation $$4 ^ { 2 x + 1 } = 8 ^ { 4 x }$$
  2. (a) Express $$3 \sqrt { 18 } - \sqrt { 32 }$$ in the form \(k \sqrt { 2 }\), where \(k\) is an integer.
    (b) Hence, or otherwise, solve $$3 \sqrt { 18 } - \sqrt { 32 } = \sqrt { n }$$
Edexcel C1 2018 June Q1
5 marks Easy -1.3
  1. (i) Simplify
$$\sqrt { 48 } - \frac { 6 } { \sqrt { 3 } }$$ Write your answer in the form \(a \sqrt { 3 }\), where \(a\) is an integer to be found.
(ii) Solve the equation $$3 ^ { 6 x - 3 } = 81$$ Write your answer as a rational number.
OCR MEI C1 Q5
3 marks Easy -1.3
5 Solve the following equations.
  1. \(\quad 2 ^ { x } = \frac { 1 } { 8 }\).
  2. \(\quad x ^ { - \frac { 1 } { 2 } } = \frac { 1 } { 4 }\)
OCR C1 Q1
3 marks Easy -1.2
  1. Find the value of \(y\) such that
$$4 ^ { y + 3 } = 8$$
OCR C1 Q1
3 marks Easy -1.2
  1. Solve the equation
$$9 ^ { x } = 3 ^ { x + 2 } .$$
OCR MEI C2 2008 January Q7
5 marks Easy -1.3
7
  1. Find \(\sum _ { k = 2 } ^ { 5 } 2 ^ { k }\).
  2. Find the value of \(n\) for which \(2 ^ { n } = \frac { 1 } { 64 }\).
  3. Sketch the curve with equation \(y = 2 ^ { x }\).
CAIE P3 2020 Specimen Q1
3 marks Moderate -0.5
1 Find the set of values of \(x\) for which \(3 \left( 2 ^ { 3 x + 1 } \right) < 8\). Give your answer in a simplified exact form.
OCR C1 2010 January Q4
7 marks Easy -1.3
4 Solve the equations
  1. \(3 ^ { m } = 81\),
  2. \(\left( 36 p ^ { 4 } \right) ^ { \frac { 1 } { 2 } } = 24\),
  3. \(5 ^ { n } \times 5 ^ { n + 4 } = 25\).
OCR H240/01 2023 June Q2
8 marks Moderate -0.8
2
    1. Show that \(\frac { 1 } { 3 - 2 \sqrt { x } } + \frac { 1 } { 3 + 2 \sqrt { x } }\) can be written in the form \(\frac { a } { b + c x }\), where \(a , b\) and \(c\) are constants to be determined.
    2. Hence solve the equation \(\frac { 1 } { 3 - 2 \sqrt { x } } + \frac { 1 } { 3 + 2 \sqrt { x } } = 2\).
  2. In this question you must show detailed reasoning. Solve the equation \(2 ^ { 2 y } - 7 \times 2 ^ { y } - 8 = 0\).
Edexcel AS Paper 1 2021 November Q2
3 marks Moderate -0.8
  1. In this question you should show all stages of your working.
Solutions relying on calculator technology are not acceptable.
Given $$\frac { 9 ^ { x - 1 } } { 3 ^ { y + 2 } } = 81$$ express \(y\) in terms of \(x\), writing your answer in simplest form.
Edexcel C1 Q1
5 marks Easy -1.2
  1. Given that \(2 ^ { x } = \frac { 1 } { \sqrt { 2 } }\) and \(2 ^ { y } = 4 \sqrt { } 2\),
    1. find the exact value of \(x\) and the exact value of \(y\),
    2. calculate the exact value of \(2 ^ { y - x }\).
    3. \(f ( x ) = \frac { \left( x ^ { 2 } - 3 \right) ^ { 2 } } { x ^ { 3 } } , x \neq 0\).
    4. Show that \(\mathrm { f } ( x ) \equiv x - 6 x ^ { - 1 } + 9 x ^ { - 3 }\).
    5. Hence, or otherwise, differentiate \(\mathrm { f } ( x )\) with respect to \(x\).
    6. The sum of an arithmetic series is \(\sum _ { r = 1 } ^ { n } ( 80 - 3 r )\).
    7. Write down the first two terms of the series.
    8. Find the common difference of the series.
    Given that \(n = 50\),
  2. find the sum of the series.
Edexcel C1 Q1
5 marks Easy -1.2
  1. Given that \(8 = 2^k\), write down the value of \(k\). [1]
  2. Given that \(4^x = 8^{2-x}\), find the value of \(x\). [4]
Edexcel C1 Q1
3 marks Moderate -0.8
Find the value of \(y\) such that $$4^{y + 3} = 8.$$ [3]
OCR C1 Q1
4 marks Moderate -0.5
Find the value of \(y\) such that $$4^{y+1} = 8^{2y-1}.$$ [4]
AQA AS Paper 1 Specimen Q3
4 marks Easy -1.3
  1. Write down the value of \(p\) and the value of \(q\) given that:
    1. \(\sqrt{3} = 3^p\) [1 mark]
    2. \(\frac{1}{9} = 3^q\) [1 mark]
  2. Find the value of \(x\) for which \(\sqrt{3} \times 3^x = \frac{1}{9}\) [2 marks]
SPS SPS FM 2024 October Q1
8 marks Moderate -0.8
    1. Show that \(\frac{1}{3-2\sqrt{x}} + \frac{1}{3+2\sqrt{x}}\) can be written in the form \(\frac{a}{b+cx}\), where \(a\), \(b\) and \(c\) are constants to be determined. [2]
    2. Hence solve the equation \(\frac{1}{3-2\sqrt{x}} + \frac{1}{3+2\sqrt{x}} = 2\). [2]
  2. In this question you must show detailed reasoning. Solve the equation \(2^{2x} - 7 \times 2^x - 8 = 0\). [4]