What is General powers? — Mathematics

Definition 4.2 University Mathematics — Year 1 · Chapter 4 — Standard Functions

For $x > 0$ and $\alpha \in \R$ : $\;x^\alpha = \eu^{\alpha \ln x}$ . For $a > 0$ , $a \neq 1$ , the logarithm to base $a$ is $\log_a x = \frac{\ln x}{\ln a}$ , the inverse of $x \mapsto a^x$ .

The growth scale of : near the right edge, x has barely passed 1.6 while x has left the frame. Every ratio (log over power, power over exponential) tends to 0 — the picture only hints at what the substitutions in the proof make exact. — The growth scale of Proposition 4.6: near the right edge, $\ln x$ has barely passed $1.6$ while $\eu^x$ has left the frame. Every ratio (log over power, power over exponential) tends to $0$ — the picture only hints at what the substitutions in the proof make exact.

Examples

Example 4.3 (Solving exponential equations)

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

x\ln2 = (x - 1)\ln5 \iff x(\ln2 - \ln5) = -\ln5 \iff x = \frac{\ln5}{\ln5 - \ln2} = \frac{\ln 5}{\ln\frac52} \approx 1.756 .

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 $(1.03)^n \geq 2$ :

n\ln(1.03) \geq \ln2 \iff n \geq \frac{\ln 2}{\ln 1.03} = \frac{0.6931}{0.02956} \approx 23.4 ,

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 $N = 2^{2026}$ :

\log_{10} 2^{2026} = 2026\,\log_{10}2 = 2026 \times 0.301030 = 609.887 ,

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).

Read in context →