Equations Involving Exponentials

Equations Involving Exponentials

A LevelAQAEdexcelOCRAQA 2022Edexcel 2022OCR 2022

Equations Involving Exponentials

Equations involving exponentials and logarithms can become much more complicated than we have already seen. On this page, we will learn how to use a calculator for logarithms, attempt to solve some more difficult equations, and apply knowledge of logarithms to real life.

The following topics build on the content in this page.

A Level AQA Edexcel OCR

Using a Calculator

Your calculator will have several buttons to do with logarithms.

\log_{\square}\square allows you to do any logarithm. Put the base in the lower box and the number in the box on the right.

\log_{10}\square is for logarithms with a base of 10 only.

\ln is the natural logarithm. We will see this later.

Example: If we want to do \log_{4}(81) on a calculator we would press \log_{\square}\square then 4 then 8 then 1 then =

A LevelAQAEdexcelOCR

Exponential Equations

Some exponential equations look more complicated, but they are really just quadratics in disguise.

They take the form ap^{2x}+bp^{x}+c=0

We solve them by substituting y=p^{x}, and noticing that y^{2}=p^{2x}

Then we have a simple quadratic: ay^{2}+by+c=0

This gives us two roots: y=r_{1} and y=r_{2}

Putting x back in gives p^{x}=r_{1} and p^{x}=r_{2}, which are two equations that we already know how to solve.

 

Example: 3\times6^{2x}-7\times6^{x}+2=0

3\times(6^{x})^{2}-7\times6^{x}+2=0

y=6^{x}

3y^{2}-7y-2=0

(3y-1)(y-2)=0

y=\dfrac{1}{3} or y=2

6^{x}=\dfrac{1}{3} or 6^{x}=2

x=\log_{6}\left(\dfrac{1}{3}\right) or x=\log_{6}(2)

x=-0.613 or x=0.387

A LevelAQAEdexcelOCR

Logarithm Equations

Difficult logarithm equations require you to use more than one law of logarithms.

Example: \log_{3}(21x-2)-2\log_{3}(x)=3

\log_{3}(21x-2)+\log_{3}(x^{-2})=3

\log_{3}((21x-2)x^{-2})=3

\log_{3}\left(\dfrac{21x-2}{x^{2}}\right)=3

\dfrac{21x-2}{x^{2}}=3^{3}

\dfrac{21x-2}{x^{2}}=27

21x-2=27x^{2}

27x^{2}-21x+2=0

(9x-1)(3x-2)=0

x=\dfrac{1}{9} and x=\dfrac{2}{3}

A LevelAQAEdexcelOCR

Real-Life Problems

Exponentials and logarithms appear frequently in real life. An important skill is being able to solve real world problems involving exponentials and logarithms.

Example: A car depreciates in value over the course of several years according to the function V=14000\times0.9^{T} where V is the value of the car and T is the time in years.

i) How much does the car cost new?

ii) Jon has owned his car for 9 years. How much is it worth?

iii) Clarissa buys a new car today. How many years until it is worth half of what she paid?

 

i) New is at T=0 so V=14000\times0.9^{0}=14000

ii) This is T=9 so V=14000\times0.9^{9}=5424

iii) She paid new cost which is 14000

Half of what she paid is 7000

7000=14000\times0.9^{T}

0.9^{T}=\dfrac{7000}{14000}

0.9^{T}=0.5

\begin{aligned}T&=\log_{0.9}(0.5)\\[1.2em]&=6.58\end{aligned}

A LevelAQAEdexcelOCR

Example Questions

3^{2x}-28\times3^{x}+27=0

(3^{x})^{2}-28\times3^{x}+27=0

 

Make a substitution y=3^{x}

y^{2}-28y+27=0

(y-27)(y-1)=0

y=27 or y=1

 

Put our substitution back in:

3^{x}=27 or 3^{x}=1

x=\log_{3}(27) or x=\log_{3}(1)

x=3 or x=0

2\log_{2}(2x-1)-3\log_{2}(x)=4

 

\log_{2}((2x-1)^{2})-\log_{2}(x^{3})=4

 

\log_{2}\left(\dfrac{(2x-1)^{2}}{x^{3}}\right)=4

 

\dfrac{(2x-1)^{2}}{x^{3}}=2^{4}

 

\dfrac{(2x-1)^{2}}{x^{3}}=16

 

(2x-1)^{2}=16x^{3}

 

4x^{2}-4x+1=16x^{3}

 

16x^{3}-4x^{2}+4x-1=0

 

(4x-1)(4x^{2}+1)=0

 

x=\dfrac{1}{4} is the only real solution.

R=1000\times0.8^{T}

 

Find initial value, which is at T=0:

 

\begin{aligned}R&=1000\times0.8^{0}\\[1.2em]&=1000\times1\\[1.2em]&=1000\end{aligned}

 

Hence, half-life is when R=\dfrac{1000}{2}=500

 

500=1000\times0.8^{T}

 

\dfrac{500}{1000}=0.8^{T}

 

0.5=0.8^{T}

 

\begin{aligned}T&=\log_{0.8}(0.5)\\[1.2em]&=3.11\text{ years}\end{aligned}

 

 

R=5

 

5=1000\times0.8^{T}

 

\dfrac{5}{1000}=0.8^{T}

 

0.005=0.8^{T}

 

\begin{aligned}T&=\log_{0.8}(0.005)\\[1.2em]&=23.7\text{ years}\end{aligned}

Related Topics

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The Exponential Function

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Formula Booklet

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