Logarithms Revision Notes for Leaving Cert Applied Maths

Logarithms

What are logarithms?

Logarithms are essentially the reverse operation of raising numbers to powers. When you're working with exponential expressions, logarithms help you find the unknown power or index.

chatImportant

The Fundamental Relationship

The core relationship between exponentials and logarithms can be written as:

$a^m = n \leftrightarrow \log\_a n = m$

In this relationship:

a is the base (must be positive and not equal to 1)
m is the power or index
n is the result
We read this as "log of n to the base a equals m"

Example in action

If $2^5 = 32$ , then using logarithms we can write: $\log\_2 32 = 5$

This means "the power you need to raise 2 to get 32 is 5."

Laws of logarithms

Understanding logarithmic laws is crucial for manipulating expressions and solving equations. There are eight fundamental laws that govern how logarithms behave mathematically.

The most essential laws for your studies include:

Law 1: Product rule

When you have multiplication inside a logarithm, you can separate it into addition of individual logarithms.

$\log\_a(xy) = \log\_a x + \log\_a y$

Law 2: Quotient rule

Division inside a logarithm becomes subtraction of separate logarithms.

$\log\_a\left(\frac{x}{y}\right) = \log\_a x - \log\_a y$

Law 3: Power rule

A power inside a logarithm can be moved to the front as a multiplier.

$\log\_a(x^p) = p \log\_a x$

Law 6: Inverse relationship

This demonstrates the inverse nature of logarithms and exponentials.

$\log\_a(a^x) = x$

infoNote

These laws provide the mathematical foundation for transforming complex logarithmic expressions into simpler, more manageable forms. Mastering these will make solving logarithmic problems much more efficient.

Simplifying logarithmic expressions

When simplifying expressions containing logarithms, you systematically apply the laws to reduce complexity. The key is working step-by-step and choosing the most efficient approach.

lightbulbExample

Worked Example: Simplifying Logarithmic Expressions

Let's simplify: $\log\_3 2 + 2 \log\_3 3 - \log\_3 18$

Approach 1: Starting by recognising that $\log\_3 3 = 1$ and breaking down $\log\_3 18$ :

$\log\_3 2 + 2(1) - \[\log\_3 9 + \log\_3 2]$
$\log\_3 2 + 2 - \log\_3 9 - \log\_3 2$
The $\log\_3 2$ terms cancel: $2 - \log\_3 9$
Since $\log\_3 9 = 2$ : $2 - 2 = 0$

Approach 2: Using the power rule first:

$\log\_3 2 + \log\_3 3^2 - \log\_3 18$
$\log\_3 2 + \log\_3 9 - \log\_3 18$
$\log\_3(2 \times 9) - \log\_3 18$ (using product rule)
$\log\_3 18 - \log\_3 18 = 0$

Both approaches yield the same result, demonstrating the flexibility these laws provide.

Solving logarithmic equations

The fundamental principle for solving logarithmic equations is straightforward and powerful:

chatImportant

The Equal Base Principle

$\text{If } \log\_a b = \log\_a c, \text{ then } b = c$

This principle works because logarithmic functions are one-to-one - each input produces exactly one output. However, the bases must be identical for this rule to apply.

lightbulbExample

Worked Example: Solving Logarithmic Equations

Solve: $\log\_3(x + 1) = \log\_3(x - 1)^2$

Since both logarithms have the same base, we can equate their arguments:

$x + 1 = (x - 1)^2$
$x + 1 = x^2 - 2x + 1$
$x^2 - 3x = 0$
$x(x - 3) = 0$
Solutions: $x = 0$ or $x = 3$

Verifying solutions

Always check your solutions by substituting back into the original equation. This step is critical because logarithms are only defined for positive arguments.

infoNote

Solution Verification Process

For $x = 0$ : $\log\_3(0 + 1) = \log\_3(0 - 1)^2$ becomes $\log\_3(1) = \log\_3(1)$ , which is valid.

For $x = 3$ : $\log\_3(3 + 1) = \log\_3(3 - 1)^2$ becomes $\log\_3(4) = \log\_3(4)$ , which is also valid.

Both solutions work in this case.

Alternative strategy - combining logarithms

Sometimes it's more efficient to combine logarithms before solving.

For equations like $\log\_3(x + 1) - \log\_3(x - 1) = 1$ :

Apply the quotient rule: $\log\_3\left(\frac{x+1}{x-1}\right) = 1$
Convert to exponential form: $\frac{x+1}{x-1} = 3^1 = 3$
Solve the resulting equation: $x + 1 = 3(x - 1)$
This gives: $x + 1 = 3x - 3$ , so $2x = 4$ , therefore $x = 2$

Remember to verify this solution works in the original equation.

Natural logarithms

When logarithms use the special mathematical constant e (approximately 2.718) as their base, we call them natural logarithms. Instead of writing $\log\_e$ , we use the notation ln.

For instance: $\ln 5$ means $\log\_e 5$

infoNote

Natural logarithms follow exactly the same laws as other logarithms. They appear frequently in calculus and many scientific applications, making them particularly important in advanced mathematics.

Exam strategies

Understanding these key strategies will improve your performance:

infoNote

Essential Exam Tips

Check domains carefully: Logarithms are only defined for positive arguments - always verify this
Substitute solutions back: This catches errors and ensures validity, especially important for logarithmic equations
Apply laws systematically: Work through one law at a time rather than attempting multiple steps simultaneously
Verify matching bases: The rule for equating arguments only works when bases are identical
Show detailed working: Clear step-by-step solutions can earn partial marks even with incorrect final answers

bookmarkSummary

Key Points to Remember:

Logarithms reverse the exponential operation: if $a^m = n$ , then $\log\_a n = m$
The three most important laws are product (becomes addition), quotient (becomes subtraction), and power (moves to front)
For equations with matching bases: if $\log\_a b = \log\_a c$ , then you can conclude $b = c$
Always verify solutions ensure all arguments remain positive
Natural logarithms ( $\ln$ ) use base $e$ and follow identical rules to other logarithms

Logarithms (Leaving Cert Applied Maths): Revision Notes

Logarithms

What are logarithms?

Example in action

Laws of logarithms

Law 1: Product rule

Law 2: Quotient rule

Law 3: Power rule

Law 6: Inverse relationship

Simplifying logarithmic expressions

Solving logarithmic equations

Verifying solutions

Alternative strategy - combining logarithms

Natural logarithms

Exam strategies

Explore Leaving Cert Applied Maths Model Answers by Topics

Calculus & Differential Equations

Real-World Applications

Differential Equations

Real-World Applications

Explore Leaving Cert Applied Maths Quizzes by Topics

Calculus & Differential Equations

Real-World Applications

Differential Equations

Real-World Applications

Explore Leaving Cert Applied Maths Flashcards by Topics

Calculus & Differential Equations

Real-World Applications

Differential Equations

Real-World Applications

Join 100,000+ Leaving Cert students studying Revision Notes with us.