Exponents, Exponentials and Logarithms · Strategy summary

Exponents, Exponentials and Logarithms: strategy summary

1. Exponents: multiplying adds exponents, dividing subtracts, and a power of a power multiplies. The outside power reaches the coefficient too.
2. Negative and fractional exponents: $x^{-n} = \frac{1}{x^n}$, so a negative exponent means a reciprocal, not a negative value; $a^{m/n}$ is the $n$th root, then the $m$th power.
3. Models: $a(1 + r)^t$ for growth, $a(1 - r)^t$ for decay, $a \cdot 2^{t/d}$ for doubling every $d$. A constant ratio in a table means exponential.
4. Logs: $\log_b x = y$ means $b^y = x$. $\log_b 1 = 0$, $\log_b b = 1$, and the input must be positive.
5. Log rules: products add, quotients subtract, powers come to the front. $\log(a + b)$ does not split.
6. Equations: match bases or take logs; condense log sums, rewrite as a power, and check every root in the original equation.