Logarithm

The inverse of Exponents.

💁 \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”

  $lo{g}_2(x)=y$ → $2^y=x$

  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.

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Source §

Related §

• Logarithms & Exponents