Number and Algebra

The laws of exponents with rational exponents allow us to simplify and manipulate expressions involving powers. These laws are fundamental in algebra and are used to solve a variety of mathematical problems. Similarly, logarithms provide a way to undo exponential functions and have their own set of laws that make calculations more straightforward. The change of base formula for logarithms is particularly useful when working with logarithms of different bases. Solving exponential equations often involves using logarithms to rewrite and solve the equation.

Laws of Exponents with Rational Exponents:

1. Product of Powers: \( a^m \times a^n = a^{m+n} \)
2. Quotient of Powers: \( \frac{a^m}{a^n} = a^{m-n} \)
3. Power of a Power: \( (a^m)^n = a^{mn} \)
4. Power of a Product: \( (ab)^n = a^n b^n \)
5. Negative Exponent: \( a^{-n} = \frac{1}{a^n} \)
6. Rational Exponent: \( a^{\frac{m}{n}} = \sqrt[n]{a^m} \)

Example: Simplify \( 2^{\frac{3}{2}} \times 2^{\frac{1}{2}} \)
Using the Product of Powers rule:
\( 2^{\frac{3}{2}} \times 2^{\frac{1}{2}} = 2^{\frac{3}{2} + \frac{1}{2}} = 2^2 = 4 \)

Laws of Logarithms:

1. Product Rule: \( \log_b(mn) = \log_b(m) + \log_b(n) \)
2. Quotient Rule: \( \log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n) \)
3. Power Rule: \( \log_b(m^n) = n \cdot \log_b(m) \)

Change of Base Formula:
For any positive numbers \( a \), \( b \), and \( c \) where \( a \neq 1 \) and \( b \neq 1 \):
\[ \log_b(c) = \frac{\log_a(c)}{\log_a(b)} \]

Example: Express \( \log_2(8) \) in terms of natural logarithm (base \( e \)):
Using the Change of Base Formula:
\[ \log_2(8) = \frac{\ln(8)}{\ln(2)} \]

Solving Exponential Equations:
To solve \( b^x = c \) for \( x \), take the logarithm of both sides:
\[ \log{(b^x)} = \log{(c)} \] \[ x\log{(b)} = \log{(c)} \] \[ x = \frac{\log(c)}{\log(b)} \]

Example: Solve \( 2^x = 8 \) for \( x \):
Taking the logarithm of both sides:
\[ x = \frac{\log(8)}{\log(2)} = \frac{\log(2^3)}{\log(2)} = \frac{3\log(2)}{\log(2)} = 3 \]

This video on Exponent Laws and Solving Equations is from Revision Village and is aimed at students taking the IB Maths AA Standard level course

This video on Exponential and Logarithmic Functions is from Revision Village and is aimed at students taking the IB Maths AA Standard level course.

This video on Log Laws and Solving Equations is from Revision Village and is aimed at students taking the IB Maths AA Standard level course