Logarithmic Properties and Change of Base

By the end of this lesson, you will consolidate logarithm properties and change-of-base at a pre-calculus level, including natural log properties and applications in more complex algebraic contexts.

Summary of Logarithm Properties

$$\log_b(MN) = \log_b M + \log_b N$$ $$\log_b!\left(\frac{M}{N}\right) = \log_b M - \log_b N$$ $$\log_b(M^p) = p\log_b M$$ $$\log_b x = \frac{\ln x}{\ln b} = \frac{\log x}{\log b}$$

Natural Logarithm Properties

$\ln(e) = 1$; $\ln(1) = 0$; $e^{\ln x} = x$; $\ln(e^x) = x$

$\ln(xy) = \ln x + \ln y$; $\ln(x/y) = \ln x - \ln y$; $\ln(x^n) = n\ln x$

Example 1 — Complex Expansion

Expand: $\ln\left(\dfrac{x^3\sqrt{y}}{e^{2z}}\right)$

$= \ln(x^3) + \ln(\sqrt{y}) - \ln(e^{2z}) = 3\ln x + \frac{1}{2}\ln y - 2z$

Example 2 — Condensing

Condense: $2\ln x - \frac{1}{3}\ln y + \ln z$

$= \ln(x^2) - \ln(y^{1/3}) + \ln z = \ln!\left(\frac{x^2 z}{y^{1/3}}\right)$

Example 3 — Change-of-Base Application

$\log_6 50 = \dfrac{\ln 50}{\ln 6} = \dfrac{3.912}{1.792} \approx 2.183$

Logarithmic Scales

Richter scale (earthquake): each unit increase = $10\times$ more ground motion.

$M = \log_{10}(I/I_0)$: magnitude 7 is $10^5$ times stronger than magnitude 2.

Decibels (sound): $dB = 10\log_{10}(I/I_0)$

pH: $pH = -\log_{10}[H^+]$

All three use the same structure: logarithm of a ratio.

Example 4 — Scientific Application

An earthquake reads $10^{4.5}$ on the seismograph compared to baseline. Find its Richter magnitude.

$M = \log_{10}(10^{4.5}) = 4.5$

Summary

All properties carry from Algebra 2 — at Pre-Calculus level, apply to more complex expressions and real-world scale models.

Natural log: preferred in calculus because $\frac{d}{dx}[\ln x] = 1/x$ (cleanest derivative).

Change of base: $\log_b x = \ln x/\ln b$ — converts to any computable base.