Properties of Logarithmic Functions

Video lesson · from 9:113 worked examples

One lesson video covers all of Exponentials & Logarithms, so it opens at 9:11 for this topic — not from the beginning.

A logarithmic function is a function of the form $f(x) = \log_a x$ where $a>0$ and $x>0$ .

The relationship between exponential functions and logarithmic functions can be expressed as:

y = \log_a x \iff a^y = x

If we sum logarithmic functions with the same base numbers then the terms can be combined by multiplying the arguments:

\log_a x + \log_a y = \log_a(xy)

If we subtract logarithmic functions with the same base numbers, then the terms can be combined by dividing the arguments:

\log_a x - \log_a y = \log_a \left(\frac{x}{y}\right)

If the argument of a logarithmic function is raised to a power, then this equates to the product of the exponent and the logarithmic function:

\log_a (x^n) = n \log_a x

The bases must match: the log laws only apply to logarithms of the same base. Terms with different bases cannot be combined at all.
$\log(a+b)$ is not $\log a + \log b$ : the addition law works the other way — a sum of logs becomes a log of a product.
Switching forms is the core skill: being unable to move between $a^y = x$ and $\log_a x = y$ is the most repeated failure in this whole topic.
Keep the argument positive: check any answer back in the original expression and reject values that make a log undefined.

Example 1

Simplify $\log_a 3 + \log_a 6$

\log_a 3 + \log_a 6 = \log_a(3 \times 6) = \log_a 18

Example 2

Simplify $\log_a 15 - \log_a 3$

\log_a 15 - \log_a 3 = \log_a\left(\frac{15}{3}\right) = \log_a 5

Example 3

Express $2\log_a 3$ in the form $\log_a x$ .

2\log_a 3 = \log_a(3^2) = \log_a 9