# 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.

A Level

## 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 Level

## 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 Level

## 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 Level

## 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 Level

## 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}

A Level

A Level

A Level

## You May Also Like...

### A Level Maths Revision Cards

The best A level maths revision cards for AQA, Edexcel, OCR, MEI and WJEC. Maths Made Easy is here to help you prepare effectively for your A Level maths exams.

£14.99

### A Level Maths – Cards & Paper Bundle

A level maths revision cards and exam papers for Edexcel. Includes 2022 predicted papers based on the advance information released in February 2022! MME is here to help you study from home with our revision cards and practise papers.

From: £22.99

### Transition Maths Cards

The transition maths cards are a perfect way to cover the higher level topics from GCSE whilst being introduced to new A level maths topics to help you prepare for year 12. Your ideal guide to getting started with A level maths!

£8.99