Properties of Logarithms

These are my notes on properties of logarithms.

Logarithms possess several important properties. One property of logarithms states that the sum of the logarithms of two numbers equals the logarithm of their product. 

\(\log 5 + \log 2 = \log 10\)

And

\( \log 4 + \log 25 = \log 100 \)

The properties of logarithms are a direct result of the inverse property:

 \(\log_a m + \log_a n = \log_a mn\)

Example 1

Expand \(\log xy\)

\(\boxed{ \log x + \log y}\)

Example 2

Expand \(ln\frac{6}{z}\)

\(\boxed{ln 6 - ln z} \)

Example 3

Expand \(\log 2x^{4}\)

\(\log 2 + \log x^{4}\)

\(\boxed{\log 2 + 4 \log x}\)

Example 4

Expand \( ln \frac{7x^{3}}{k} \)

\( ln 7x^{3} - ln k \)

\( ln 7 + ln x^{3} - ln k \)

\( \boxed{ln 7 + 3 ln x - ln k} \)

Example 5

Combine expressions: \(ln 2e + ln \frac{1}{e} \)

Factor out \(ln\)

\( ln (2e * \frac{1}{e}) \)

\(\boxed{ ln 2} \)

Example 6

Combine expressions: \( \log_2 27 + \log_2 x^{3} \)

\( \boxed{\log_2(27x^{3})} \)

Example 7

Combine expressions: \( \log x^{3} - \log x^{2} \)

\( = \log\frac{x^{3}}{x^{2}} \)

\(\boxed{\log x} \)

Example 8

Make these a single expression:

\( \log 5 + \log 15 - \log 3 \)

\( \log(5*15) - \log 3 \)

\( \log 75 - \log 3 \)

\( \log \frac{75}{3} \)

\( \boxed{ \log 25} \)