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AQA A-level Maths Exponentials

Exponentials

This page covers the following topics:

An exponential function such as aΛ£ = b can be solved by rewriting it as logab = x, where a is the base of the logarithm. The logarithm can now be inputted on a calculator to solve for x. If the base of the logarithm is not written, it should be assumed to be 10.

Basics of exponentials

The natural exponential function is y = eΛ£, where e is Euler's number. The inverse of the natural exponent is the natural logarithm, lnx. This can be used to solve equations with the two functions.

The exponential function

All laws of indices apply when using the natural exponential function: when multiplying, the indices are added, and when dividing, the indices are subtracted. When Euler's number is raised to the power of 0, it equates to 1.

Laws of indices

Equations involving eΛ£ can be solved using the laws of indices and the knowledge that the natural logarithm is its inverse.

Solving equations with exponents

1

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Solve 5Λ£ = 3125.

AQA A-level Maths Exponentials Solve 5Λ£ = 3125.
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2

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Find the value of x in logβ‚‚(4x) = 4.

AQA A-level Maths Exponentials Find the value of x in logβ‚‚(4x) = 4.
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3

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Calculate, to 2 decimal points, eΛ£ for when x = 5.

AQA A-level Maths Exponentials Calculate, to 2 decimal points, eΛ£ for when x = 5.
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4

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Given that 1/2 Γ— lnx + 6 = 8, find the value of x to 3 significant figures.

AQA A-level Maths Exponentials Given that 1/2 Γ— lnx + 6 = 8, find the value of x to 3 significant figures.
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5

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Find the unknown value of y-coordinate of the point given on the diagram. Give the answer in 3 significant figures.

AQA A-level Maths Exponentials Find the unknown value of y-coordinate of the point given on the diagram. Give the answer in 3 significant figures.
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6

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Simplify e⁡ Γ— e⁢/eΒ².

AQA A-level Maths Exponentials Simplify e⁡ Γ— e⁢/eΒ².
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7

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Give the following expression in simplified form: e⁡ Γ— 15e⁴ Γ— 2e/3e⁡.

AQA A-level Maths Exponentials Give the following expression in simplified form: e⁡ Γ— 15e⁴ Γ— 2e/3e⁡.
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8

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The graph of y = 3eΒ²Λ£ Γ— eΛ£ is provided. Calculate the y-coordinate of the point A to 3 significant figures.

AQA A-level Maths Exponentials The graph of y = 3eΒ²Λ£ Γ— eΛ£ is provided. Calculate the y-coordinate of the point A to 3 significant figures.
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9

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Solve 20eΒ³Λ£ βˆ’ 8 = 11eΒ³Λ£ + 19 for x.

AQA A-level Maths Exponentials Solve 20eΒ³Λ£ βˆ’ 8 = 11eΒ³Λ£ + 19 for x.
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10

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Given that (eΛ£ βˆ’ 8)(e⁡ˣ βˆ’ 3) = 0, calculate the possible values of x.

AQA A-level Maths Exponentials Given that (eΛ£ βˆ’ 8)(e⁡ˣ βˆ’ 3) = 0, calculate the possible values of x.
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