Title: InkMaths / Indices and log laws
On every note of this deck, unless a note says otherwise:
Footer: Continue on InkMaths · © 2026 InkMaths. All rights reserved.

Front: STANDARD RESULT · LEARN IT
Exponential and log undo each other:
e^ln x = ?, ln(e^x) = ?\(e^{\text{ln }x} = ?, \text{ln}(e^{x}) = ?\)e^ln x = ?, ln(e^x) = ?
Back: x for x > 0, x for every x\(x \text{ for } x > 0, x \text{ for every } x\)x for x > 0, x for every x

Front: STANDARD RESULT · LEARN IT
Quotient law:
log_a x − log_a y = ?\(\text{log}_{a} x − \text{log}_{a} y = ?\)log_a x − log_a y = ?
Back: log_a x − log_a y = log_a xy\(\text{log}_{a} x − \text{log}_{a} y = \text{log}_{a} \frac{x}{y}\)log_a x − log_a y = log_a (x/y)

Front: STANDARD RESULT · LEARN IT
Power law:
k log_a x = ?\(k \text{log}_{a} x = ?\)k log_a x = ?
Back: k log_a x = log_a (x^k)\(k \text{log}_{a} x = \text{log}_{a} (x^{k})\)k log_a x = log_a (x^k)

Front: RECALL
State how to straighten an exponential or power model.
Back: For y = ax to the n, take logs of both sides to get log y = n log x + log a, so plot log y against log x. Gradient n, intercept log a. For y = ab to the x, plot log y against x instead. Gradient log b, intercept log a. Which axis gets the log is the whole question.

Front: SPOT THE ERROR
Say what is wrong with this claim:
log(x + y) is the same as log x + log y.
Back: The product law says log x + log y = log(xy). A log turns a product into a sum. It does nothing at all to a sum, so log(x + y) stays as it is. Try x = y = 10. log 20 is about 1.3, and log 10 + log 10 = 2.

Front: CHECK YOURSELF
Write 2 log 3 + log 4 as a single logarithm.
Back: 2 log 3 = log 9, and log 9 + log 4 = log 36.

Front: STANDARD RESULT · LEARN IT
Product law:
log_a x + log_a y = ?\(\text{log}_{a} x + \text{log}_{a} y = ?\)log_a x + log_a y = ?
Back: log_a x + log_a y = log_a (xy)\(\text{log}_{a} x + \text{log}_{a} y = \text{log}_{a} (xy)\)log_a x + log_a y = log_a (xy)

Front: DEFINE
Define: logarithm.
Back: The power to which a must be raised to give x. So a^y = x and y = log base a of x say the same thing, with a > 0, a ≠ 1 and x > 0.

Front: STANDARD RESULT · LEARN IT
Negative and fractional indices:
a^-n = ?, a^m/n = ?, a^0 = ?\(a^{-n} = ?, a^{m/n} = ?, a^{0} = ?\)a^(-n) = ?, a^(m/n) = ?, a^0 = ?
Back: 1a^n, the nth root of a, raised to m, 1\(\frac{1}{a^{n}}, \text{the nth root of a, raised to m}, 1\)(1/a^n), the nth root of a, raised to m, 1

Front: STANDARD RESULT · LEARN IT
Laws of indices:
a^ma^n = ?, a^ma^n = ?, (a^m)^n = ?\(a^{m}a^{n} = ?, \frac{a^{m}}{a^{n}} = ?, (a^{m})^{n} = ?\)a^ma^n = ?, (a^m/a^n) = ?, (a^m)^n = ?
Back: a^m+n, a^m-n, a^mn\(a^{m+n}, a^{m-n}, a^{mn}\)a^(m + n), a^(m - n), a^mn