Edexcel IAL - Pure Maths 2- 5.3 Solving Equations of the Form ax = b- Study notes - New syllabus
Edexcel IAL – Pure Maths 2- 5.3 Solving Equations of the Form ax = b -Study notes- New syllabus
Edexcel IAL – Pure Maths 2- 5.3 Solving Equations of the Form ax = b -Study notes -Edexcel A level Physics – per latest Syllabus.
Key Concepts:
- 5.3 Solving Equations of the Form ax = b
Solving Exponential Equations of the Form \( a^x = b \)
An equation of the form \( a^x = b \), where \( a > 0,\ a \neq 1 \) and \( b > 0 \), is called an exponential equation.
To solve such equations, logarithms are used.
Using Logarithms
Starting from:
\( a^x = b \)
Take logarithms on both sides:
\( \log(a^x) = \log b \)
Using the power rule of logarithms:
\( x \log a = \log b \)
Hence:
\( x = \dfrac{\log b}{\log a} \)
Change of Base Formula
If the logarithm \( \log a \) is not directly available, we use the change of base formula:
\( \log_a b = \dfrac{\log b}{\log a} \)
Any base may be used, commonly base 10 or base \( e \).
Important Conditions
- \( a > 0 \) and \( a \neq 1 \)
- \( b > 0 \)
- Solutions may be exact or decimal
Example
Solve the equation:
\( 2^x = 8 \)
▶️ Answer / Explanation
Write 8 as a power of 2:
\( 8 = 2^3 \)
So:
\( 2^x = 2^3 \Rightarrow x = 3 \)
Solution: \( x = 3 \)
Example
Solve the equation:
\( 5^x = 7 \)
▶️ Answer / Explanation
Take logarithms on both sides:
\( x \log 5 = \log 7 \)
So:
\( x = \dfrac{\log 7}{\log 5} \)
Using a calculator:
\( x \approx 1.209 \)
Solution: \( x \approx 1.21 \)
Example
Solve the equation:
\( 3^{2x-1} = 10 \)
▶️ Answer / Explanation
Take logarithms:
\( (2x – 1)\log 3 = \log 10 \)
Since \( \log 10 = 1 \):
\( 2x – 1 = \dfrac{1}{\log 3} \)
\( 2x = 1 + \dfrac{1}{\log 3} \)
\( x = \dfrac{1}{2}\!\left(1 + \dfrac{1}{\log 3}\right) \)
Using a calculator:
\( x \approx 1.548 \)
Solution: \( x \approx 1.55 \)