Leaving Cert Logs & Indices (Higher Level): Laws of Logs, Surds & Exponentials

Logs and indices cover the laws of indices, surds, logarithms and solving exponential equations. The topic appears on Paper 1 and supports financial maths and calculus.

Written by Sean Doyle · Last updated: 2026-06-12

Key facts

Logarithms are the inverse of exponentials: if 10^x = y then log₁₀(y) = x.
The laws of indices and laws of logarithms are provided in the formula and tables booklet.
Exponential equations are usually solved by taking logs of both sides.
Logs and indices are examined on Paper 1.

Logs and Indices explained

What are Indices?

Index notation (or exponential notation) is a shorthand way to write repeated multiplication.

$a^n = a \times a \times a \times \ldots \times a \text{ (n times)}$

Where: - $a$ = base - $n$ = index or exponent or power

Examples: - $2^4 = 2 \times 2 \times 2 \times 2 = 16$ - $5^3 = 5 \times 5 \times 5 = 125$

Special Cases

Important index values:

$a^1 = a$ (anything to power 1 equals itself)

$a^0 = 1$ (anything to power 0 equals 1, where $a \neq 0$ )

$a^{-n} = \frac{1}{a^n}$ (negative indices mean reciprocals)

Example: $2^{-3} = \frac{1}{2^3} = \frac{1}{8}$

Exponential Functions

An exponential function has the form: $f(x) = a^x$

Where $a > 0$ and $a \neq 1$ (the base is constant, the exponent is the variable).

The Natural Exponential Function: $f(x) = e^x$

Where $e \approx 2.71828$ is Euler's number (a special mathematical constant).

Exponential functions model growth and decay in nature, finance, and science.

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Key formulas

Name	Formula	Description
Index Law (Multiply)	$a^m \times a^n = a^{m+n}$	Add powers when multiplying same base
Index Law (Divide)	$a^m \div a^n = a^{m-n}$	Subtract powers when dividing same base
Index Law (Power)	$(a^m)^n = a^{mn}$	Multiply powers when raising power to power
Zero Index	$a^0 = 1$	Any number to power zero equals one
Negative Index	$a^{-n} = \frac{1}{a^n}$	Negative power means reciprocal
Fractional Index	$a^{\frac{m}{n}} = \sqrt[n]{a^m}$	Fractional power as root

Worked examples

Worked example 1

If $\log_3 81 = x$ , what is $x$ ?

Since $3^4 = 81$ , we have $\log_3 81 = 4$

Answer:

Worked example 2

What is $\ln e^3$ ?

Using $\ln(e^x) = x$ : $\ln e^3 = 3$

Answer:

Where students lose marks

Mixing up the laws of logs (e.g. log(a) + log(b) = log(ab), not log(a+b)).
Forgetting to rationalise the denominator when simplifying surds.
Errors with negative and fractional indices.

Frequently asked questions

What are the laws of logarithms?

The three core laws let you turn products into sums, quotients into differences, and powers into multipliers. They are listed in the exam formula and tables booklet.

How do I solve an exponential equation?

Take the logarithm of both sides, then use the power law to bring the unknown exponent down so you can solve for it.

Are logs and indices on Paper 1 or Paper 2?

Logs and indices are examined on Paper 1.

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