Indices and Logarithm Rules

NCEA Level 3 Calculus 2026 Complex Numbers

When Multiplying Numbers

log_b(xy)

Add the Logs

log_bx + log_by

When Dividing Numbers

log_b(x/y)

Subtract the Logs

log_bx - log_by

When Raising a Number to a Power

log_b(xⁿ)

Multiply the Log by that Number

nlog_bx

Multiplication of Powers

y^m x yⁿ

Add the Powers

y^m+n

Division of Powers

y^m/yⁿ

Subtract the Powers

y^m-n

Raising to a Power

(y^m)ⁿ

Multiply the Powers

y^m.n

Negative Powers

y^-m

Positive Denominator

1/y^m

Fractional Powers

y^1/m or y^n/m

Find the appropriate root/surd

m (root) y or m (to the root) yⁿ

Exponential Function Definition

A function where the variable is an exponent

Indices Definition

Plural of the ‘index’ or ‘power’ of an exponential function

Logarithms Definition

Inverse of an exponential function

The ‘log’ of a number

The power to which the base must be raised to give the number

The exponent of an exponential function

The ‘power’ of the ‘index’

Exponential function of e

y = e^x or f(x) = e^x

The Natural Logarithm

y = log_ex or y = lnx

Log rule applied to exponential equation to move the exponent

log_b(xⁿ) = nlog_bx