General powers কী?

Example 4.3 (Solving exponential equations)

Solve $2^x = 5^{\,x-1}$ in $\R$. Both sides are positive, so take logarithms — a reversible step:

Next, solve $x^{\sqrt2} = 3$ for $x > 0$: raise to the power $\frac1{\sqrt2}$ (i.e. apply the reciprocal bijection): $x = 3^{1/\sqrt2} = \eu^{(\ln 3)/\sqrt2} \approx 2.175$. The insight: every equation mixing powers unwinds through $\ln$ and $\exp$, because the definition $x^\alpha = \eu^{\alpha\ln x}$ reduces all power manipulations to arithmetic of exponents — but only on the domain $x > 0$ where that definition lives.

Example 4.4 (Doubling times)

A quantity grows by $3\%$ per step: after $n$ steps it is multiplied by $(1.03)^n$. When does it double? Solve $(1.03)^n \geq 2$:

so the first doubling occurs at step $24$. (The financiers’ “rule of $72$”, which estimates the doubling time as $72$ divided by the rate in percent, is this computation with the approximation $\ln(1 + x) \approx x$, quantified in Chapter 16.) Exponential processes are best reasoned about through their logarithms: on that scale, growth is linear and questions become divisions.

Example 4.5 (How long is $2^{2026}$?)

The number of decimal digits of an integer $N \geq 1$ is $\floor{\log_{10} N} + 1$ (indeed $N$ has $d$ digits exactly when $10^{d-1} \leq N < 10^d$, i.e. $d - 1 \leq \log_{10}N < d$). For $N = 2^{2026}$:

so $2^{2026}$ has $610$ digits. The fractional part carries a bonus: $10^{0.887} \approx 7.7$, so the number begins with $7$. One multiplication answered a question about a number no one will ever write out — logarithms compress multiplicative size into additive size, which is their whole historical point (Example 7.12 makes that sentence precise).