Logarithmic Function Manipulation

Exponent rules have twins for logarithms. Because a logarithm is an exponent, every rule for combining exponents becomes a rule for combining logarithms. This page states the three properties, the change of base formula, what each one means for a graph, and the domain trap that costs points.

Three properties of logarithms

For a base $b > 0$, $b \ne 1$, and positive numbers $x$ and $y$:

Why the product property holds: write $x = b^m$ and $y = b^n$, so $m = \log_b x$ and $n = \log_b y$. Then $xy = b^{m + n}$, so $\log_b(xy) = m + n$.

There is no rule for $\log_b(x + y)$, and $\log_b \frac{x}{y}$ is not $\frac{\log_b x}{\log_b y}$. Those two slips are the most common log errors on the exam.

Expanding and condensing

"A single logarithm" means exactly one log term. Stopping at $\log(3x^2) - \log 5$ does not earn the point.

Change of base

Any logarithm can be written through another base $a$:

$$\log_b x = \frac{\log_a x}{\log_a b}.$$

With a calculator, use $a = e$ or $a = 10$: $\log_7 20$$= \frac{\ln 20}{\ln 7}$$\approx 1.540$. Check the size: $7 < 20 < 49$, so the answer must be between $1$ and $2$.

The formula also says something about graphs. $\frac{1}{\log_a b}$ is a constant, so $\log_b x$ is a constant times $\log_a x$. For example, $\log_4 x$$= \frac{\log_2 x}{\log_2 4}$$= \frac{1}{2}\log_2 x$. Every logarithmic function is a vertical dilation of every other one.

What the properties do to graphs

The product property turns a horizontal dilation into a vertical translation. For $h(x) = \log_2(4x)$:

$$h(x) = \log_2 4 + \log_2 x = 2 + \log_2 x.$$

So multiplying every input by $4$ (a horizontal dilation by a factor of $\frac{1}{4}$) gives the same graph as moving $\log_2 x$ up $2$ units. The power property turns a power on the input into a vertical dilation: $\log_2(x^3) = 3\log_2 x$ for $x > 0$.

The domain trap

Each property holds only where every logarithm in it is defined. Look at $\ln(x^2)$ and $2\ln x$.

When a problem asks for an expression equal to another "for every $x$ in its domain", check a negative input before you use the power property.