Guided Example #2 - Laws of Logarithms

Logarithms with base 10 are called common logarithms. It is standard practice to omit the subscript 10 and simply write log x. There are three laws of logarithms which you must know.

Example 1

Simplify $\log_{5} 2 + \log_{5} 4$ $\log_52\:+\:\log_54$

$= \log_{5} (2 \cdot 4)$ $=\log_5\left(2\cdot4\right)$

$= \log_{5} 8$ $=\log_58$

Rule 1: $LaTeX: \log_ax+\log_ay=\log_a\left(xy\right)$

$\log_ax+\log_ay=\log_a\left(xy\right)$ where a, x, y > 0
If two logarithmic terms with the same base number (a above) are being added together, then the terms can be combined by multiplying the arguments (x and y above).

Example 2

Evaluate $LaTeX: \log_46-\log_43$

$\log_46-\log_43$

{log}_{4} 6 - {log}_{4} 3

\log_{4} 6 - \log_{4} 3

$\log_46-\log_43$

= \log_{4} (\frac{6}{3})

$=\log_4\left(\frac{6}{3}\right)$

= \log_{4} (2)

$=\log_4\left(2\right)$

If we let

x = \log_{4} 2

$x=\log_42$ , written in exponential form we have

4^{x} = 2 ⟹ 2^{2} x = 2^{1} ⟹ x = \frac{1}{2}

$4^x=2\:\Longrightarrow\:2^2x\:=\:2^1\:\Longrightarrow\:x=\frac{1}{2}$

Rule 2: $LaTeX: \log_ax-\log_ay=\log_a\left(\frac{x}{y}\right)$

$\log_ax-\log_ay=\log_a\left(\frac{x}{y}\right)$ where a, x, y > 0
If a logarithmic term is being subtracted from another logarithmic term with the same base number (a above), then the terms can be combined by dividing the arguments (x and y in this case). Note that the argument which is being subtracted (y above) appears in the denominator of the fraction when the two terms are combined.

Example 3

Express $LaTeX: 2\log_73$

$2\log_73$ in the form $LaTeX: \log_7a$

$\log_7a$

2 \log_{7} 3

$2\log_73$

= \log_{7} 3^{2}

$=\log_73^2$

\log_{7} 9

$\log_79$

Rule 3: $LaTeX: \log_ax^n$

$\log_ax^n$ where a, x > 0 .
The power of the argument (n above) can be move to the front of the term as a multiplier, and vice-versa.