Exponential function manipulation

For a positive base b and any real exponents m and n:

1. Product property: $b^m \cdot b^n = b^{m+n}$. Multiplying powers adds exponents. The base does not change, and nothing here applies to $b^m + b^n$, which has no simplification rule at all.
2. Power property: $(b^m)^n = b^{mn}$. A power of a power multiplies exponents. Note the contrast with the product rule: $(2^x)^2 = 2^{2x}$, while $2^x \cdot 2^x$ is also $2^{2x}$, but $2^x + 2^x = 2^{x+1}$.
3. Negative and fractional exponents: $b^{-n} = 1/b^n$ (a reciprocal, never a negative), and $b^{1/k}$ is the k-th root of b, so $b^{1/2} = \sqrt{b}$.

The popular wrong versions, so you recognize them on sight: multiplying the bases ($2^x \cdot 2^3$ is not $4^{x+3}$), multiplying exponents in a product ($2^x \cdot 2^3$ is not $2^{3x}$), and halving instead of rooting ($b^{1/2}$ is not $b/2$).

The product property, read backward, splits a shifted exponent: $b^{x+c} = b^c \cdot b^x$. So $5^{x+2} = 5^2 \cdot 5^x = 25 \cdot 5^x$, and in general $a \cdot b^{x+c} = (a b^c) \cdot b^x$: still an exponential in x, with the same base, but a new front coefficient.

Geometrically this is a small miracle: shifting an exponential horizontally is the same as stretching it vertically. Move $2^x$ left by 3 and you get $2^{x+3} = 8 \cdot 2^x$, the original graph magnified eight-fold. No other function family in this course collapses two different-looking transformations into one this way, and the exam likes to ask for exactly this rewrite.

Suppose a quantity doubles every week: $f(x) = a \cdot 2^x$ with x in weeks. What is the daily factor? Rewrite the clock: with t in days, $x = t/7$, so $f = a \cdot 2^{t/7} = a \cdot (2^{1/7})^t$. The daily factor is $2^{1/7} \approx 1.104$, about 10.4% growth per day.

The rule is always the same: to convert a factor across time scales, take the appropriate root or power of it, never divide it. Splitting a weekly doubling into seven days of $2/7$ each is the additive instinct sneaking back in; seven days of factor $2^{1/7}$ compound, by the power property, to exactly $2$.

Equivalent exponential forms are not interchangeable in practice; each surfaces a different quantity. $100 \cdot 1.5^{x-2}$ makes it easy to see the value 100 occurs at x = 2. Its regrouped twin $(100/1.5^2) \cdot 1.5^x \approx 44.4 \cdot 1.5^x$ surfaces the value at x = 0 instead. Same function, different facts on display.

The manipulation habit to build: before computing, ask what the question wants (initial value? factor per step? factor per decade?), then pick the rewrite that puts that number in plain sight. The rules are the vehicle; the destination is always a quantity with a meaning.