Exponential Equations

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الشرح

🦠 What you will learn

Bacteria reproduce by binary fission, doubling every hour. Starting from one cell, after how many hours are there $1024$ cells? The unknown sits in the exponent itself — and that is what an exponential equation solves.

Exponential equation

An equation with a variable in the exponent, such as

2^x=8

and

3^{x-4}=\dfrac{1}{9}

Not exponential equations

x^2=9

and

x-7=3^2

The form studied this year

We work with the form

a^{x}=b

where

a>0

and

a\neq1

Equal powers property

If two positive bases other than one are equal, their exponents are equal.

a^{x}=a^{y}\ \Rightarrow\ x=y\qquad (a>0,\ a\neq1)

How to solve

Match the bases on both sides — factorising if you have to — then equate the exponents and solve the equation you are left with.

Factorising helps

125=5^3

16=2^4

and

\dfrac{1}{9}=3^{-2}

🧩 Worked examples

أمثلة محلولة

Worked exampleBases already match

Solve $3^x=3^4$ .

Worked exampleRewrite the other side

Solve $5^x=125$ .

Worked exampleA compound exponent

Solve $2^{x-1}=16$ .

Worked exampleA fraction on the right

Solve $3^{x}=\dfrac{1}{9}$ .