**The inverse of Exponents.** > [!tldr] 💁 $\log_z(x)=y$ → $z^y=x$ # Logarithm Properties All of these properties hold for any base number. ## Log of Multiplication Multiplication within a Log turns into addition of two logs. $ \log_a(x\times y) = \log_a(x) + \log_a(y) $ ## Log of Division Division within a Log turns into subtraction of two logs. $ \log_a(\frac{x}{y}) = \log_a(x) - \log_a(y) $ ## Log of Exponentiation Exponents within a Log can be moved outside the log as coefficients. $ \log_a(x^y)=y\times \log_a(x) $ ## Change-of-Base You can do this, for any number "b": $ \log_a(x) = \frac{\log_b(x)}{\log_b(a)} $ So, if you want $\log_3(x)$ and your calculator only has buttons for "LOG" and "LN", you can just use: $\frac{\ln(x)}{\ln(3)}$. $ \log_8(15) = \frac{\log_2(15)}{\log_2(8)} \approx 1.3023 $ # Special Logarithms - Log Base 10: "The Common Logarithm" $\log(x)=y$ → $10^y=x$ "Common" because it was commonly used before calculators as a way to turn division into subtraction. With a slide rule & a table of the answers to $\log(x)$ you could figure out $\log_x(y)$ for any x or y. - Log Base _e_: "The Natural Logarithm" $\ln(x)=y$ → $e^y=x$ The Natural Logarithm is related to continuously compounding growth. If you have an investment with 5% annually compounding interest, how long will it take to double? $\ln(2)=0.693={rate}\times{time}$ $0.693 = 0.05 \times {time}$ $\frac{0.693}{0.05} = {time} = 13.86$ years - Log Base 2a: "The Binary Logarithm" $log_2(x)=y$ → $2^y=x$ Useful in knowing the number of digits necessary to representing a given number in binary, or how many layers a head-to-head bracket will require. **** ## Source ## Related - [[Logarithms & Exponents]]