SL 1.5—Integer exponents and logarithms

Use exponent laws and logarithms to solve exponential equations.

Exponent laws simplify powers; a logarithm reverses exponentiation, so aˣ=b is equivalent to x=logₐb for a>0,a≠1.

Worked example
2ˣ=16 gives x=4; 10ˣ=3 gives x=log(3).

Worked example
What operation undoes a power? logarithm, with a valid positive base and argument.

Common boundary
A logarithm is not an ordinary division or a power of ten only.

Core integer exponent laws, for non-zero $a$ where required, are $a^ma^n=a^{m+n}$, $a^m/a^n=a^{m-n}$, $(a^m)^n=a^{mn}$, $a^0=1$ and $a^{-n}=1/a^n$. For example, $5^3\cdot5^{-6}=5^{-3}=1/125$. Logarithms reverse exponentiation: $10^x=b\iff x=\log_{10}b$ and $e^x=b\iff x=\ln b$, with $b>0$.