What is a logarithm anyway?

A logarithm is just a way of asking an exponential question in reverse.

Instead of wondering “What is 2 to the power of 3?”, you ask “2 raised to what power gives me 8?”

That’s the idea behind a logarithm: it tells you the exponent.

In math language, if $a^y = x$, then $\log_a x = y$. So when you see something like $\log_2 8 = 3$, that’s because $2^3 = 8$.

Logarithms and exponentials work as opposites. For example, $a^{\log_a x} = x$, and $\log_a(a^y) = y$. On a graph, they’re reflections of each other across the line $y = x$.

Why the Restrictions Exist

Logarithms come with a few rules: the base has to be positive, it can’t equal 1, and the input must be greater than zero. These conditions come from how exponential functions behave.

If the base is negative, then raising it to real exponents gives unpredictable results. If the base is 1, then $1^x$ is always just 1, no matter what $x$ is. And if you try to take the log of zero or a negative number, there’s no real exponent that makes a positive base equal those values. That’s why $\log_a x$ only makes sense when $x$ is positive and $a > 0$, with $a \ne 1$.

The Famous Bases: log and ln

There are two types of logarithms you’ll run into most often.

The common logarithm is written $\log_{10} x$. Since our number system is base 10, this version is useful in science and engineering. For example, $\log_{10}(1000) = 3$ because $10^3 = 1000$.

The natural logarithm is written $\ln x$, which is shorthand for $\log_e x$, where $e \approx 2.718$. This number shows up in many systems that involve continuous growth or change over time.

If you invested $1 at 100% annual interest, compounded constantly, you would end up with exactly $e$ dollars after one year.

Natural logs appear in models of population growth, radioactive decay, and many other areas where the rate of change depends on the current amount.

How Logarithm Rules Work

The rules for working with logs come from the properties of exponents.

When you multiply two values inside a logarithm, the result equals the sum of the individual logs:

$\log_a(MN) = \log_a M + \log_a N$.

If you divide one value by another inside a log, the result becomes a subtraction:

$\log_a(M/N) = \log_a M – \log_a N$.

When you raise a value inside the log to a power, that exponent can move in front:

$\log_a(M^k) = k \log_a M$.

If you need to switch between different bases, you can use the change of base formula:

$\log_a M = \frac{\log_b M}{\log_b a}$. This is especially helpful when your calculator only supports $\log$ and $\ln$.

There are also a few important values to remember. $\log_a 1 = 0$, since $a^0 = 1$. $\log_a a = 1$, because $a^1 = a$. And if you take the log of a power, such as $\log_a(a^k)$, you get $k$ as the result.

Graphing Logarithmic Functions$y = \log_a x$

The graph of a logarithmic function only exists for positive $x$-values. The domain is $x > 0$, while the range covers all real numbers. It passes through the point $(1, 0)$, because the log of 1 is always zero. There’s also a vertical asymptote at $x = 0$, which means the graph approaches the y-axis but never touches it.

The base determines the shape. If the base is greater than 1, the graph rises slowly as $x$ increases. If the base is between 0 and 1, the graph falls instead. Either way, near $x = 0$, the curve becomes very steep.

You can shift or stretch the graph using the general form
$y = a \cdot \log_b(x – h) + k$. These transformations work the same way as with other parent functions.

Solving Logarithmic Equations

To solve a log equation with a single logarithm, you can rewrite it in exponential form. For example, $\log_3(x – 2) = 4$ becomes $x – 2 = 3^4 = 81$, so $x = 83$.

When you see two or more logs on the same side with the same base, combine them before solving.

For example:
$\log_2 x + \log_2(x – 3) = 3$ becomes
$\log_2(x(x – 3)) = 3$, so $x(x – 3) = 8$, and then you solve the quadratic.

If there are logs on both sides of the equation with the same base, you can just set the inputs equal to each other.

For example:
$\log_5(3x – 1) = \log_5(2x + 6)$ leads to $3x – 1 = 2x + 6$.

For exponential equations, logs are useful for bringing the exponent down.

Consider $7^{2x – 5} = 13$. You can apply a natural log to both sides:
$(2x – 5) \ln 7 = \ln 13$, then isolate $x$.

After solving, always plug your solutions back in to check. If any input to a log becomes zero or negative, that solution must be rejected.

Real-Life Applications

Logarithms appear in many real-world settings that involve rapid changes in size, scale, or intensity.

In chemistry, the pH scale measures acidity using a logarithmic formula:
$\text{pH} = -\log_{10}[H^+]$. Each unit change reflects a tenfold difference in hydrogen ion concentration.

The Richter scale for earthquakes is also logarithmic. A magnitude 6 earthquake is ten times stronger than a magnitude 5.

Decibels work in the same way. The formula
$L = 10 \log_{10}(I/I_0)$ shows how sound intensity scales by powers of 10.

You’ll also use logs in radioactive decay problems, half-life calculations, and continuously compounding interest, like in the formula
$A = Pe^{rt}$, where natural logs help you solve for time or rate.