What is a logarithm in simple terms?

A logarithm is an exponent. The statement log_b x = y means b^y = x, so log_b x is the power you raise the base b to in order to get x. For example, log₂ 8 = 3 because 2^3 = 8.

How do you convert between logarithmic and exponential form?

Use the definition log_b x = y is the same as b^y = x. The base of the log becomes the base of the power, the value of the log becomes the exponent, and the argument becomes the result. Keep the base fixed and move the exponent.

What are the main logarithm rules on the ACT?

The product rule turns log_b(xy) into log_b x + log_b y, the quotient rule turns a division into subtraction, and the power rule brings an exponent out front: log_b(x^k) = k·log_b x. Also, log_b 1 = 0 and log_b b = 1.

Why can you not take the log of a negative number?

A logarithm asks for the exponent that produces the argument from a positive base. Since a positive base raised to any real power is always positive, no exponent can produce a negative or zero result, so those logs are undefined.

CramClub • Logarithms — ACT® Guide & Practice

$\log_b x = y \iff b^y = x$

$\log_b(xy) = \log_b x + \log_b y$

$\log_b\!\left(\dfrac{x}{y}\right) = \log_b x - \log_b y$

$\log_b(x^k) = k \log_b x$

$\log_b x = \dfrac{\log x}{\log b}$

A Logarithm Is Just an Exponent

A logarithm answers the question "what exponent do I need?" The statement

$\log_b x = y \quad \text{means} \quad b^y = x$

reads as "the log, base $b$ , of $x$ equals $y$ ," and it is simply the exponent $y$ that turns the base $b$ into the number $x$ . For example, $\log_2 8 = 3$ because $2^3 = 8$ . Every logarithm question on the ACT becomes easy once you can flip fluently between logarithmic form and exponential form. The ACT typically includes one or two log questions, and they reward this single conversion skill more than anything else.

Converting Between Forms

The base stays the base; the log's value is the exponent. Practice reading each direction:

Logarithmic form	Exponential form	Why
$\log_3 81 = 4$	$3^4 = 81$	3 to the 4th is 81
$\log_5 25 = 2$	$5^2 = 25$	5 squared is 25
$\log_{10} 1000 = 3$	$10^3 = 1000$	base-10 "common log"

When no base is written, $\log x$ means $\log_{10} x$ (the common log), and $\ln x$ means $\log_e x$ (the natural log, base $e \approx 2.718$ ).

Evaluating a Logarithm

To evaluate $\log_b x$ , ask " $b$ raised to what power gives $x$ ?" It helps to rewrite $x$ as a power of $b$ .

Evaluate $\log_2 32$ . Since $32 = 2^5$ , we have $\log_2 32 = 5$ .

Evaluate $\log_9 3$ . We need $9^{?} = 3$ . Because $9^{1/2} = 3$ , the answer is $\tfrac{1}{2}$ . Fractional answers are common when the number is a root of the base.

The Log Rules

Logs turn multiplication into addition, which is exactly why they were invented. Three rules cover the ACT:

$\log_b(xy) = \log_b x + \log_b y$ $\log_b\!\left(\frac{x}{y}\right) = \log_b x - \log_b y$ $\log_b(x^k) = k \log_b x$

The product of arguments becomes a sum of logs, a quotient becomes a difference, and an exponent comes out in front as a multiplier. Two facts round out the toolkit: $\log_b 1 = 0$ (because $b^0 = 1$ ) and $\log_b b = 1$ (because $b^1 = b$ ).

Combining and Expanding

The rules run both directions. To combine $\log_2 5 + \log_2 3$ , use the product rule: $\log_2(5 \cdot 3) = \log_2 15$ . To expand $\log_3\!\left(\frac{x^2}{y}\right)$ , use the quotient and power rules: $2\log_3 x - \log_3 y$ .

Solving a Simple Log Equation

Most ACT log equations are solved by converting to exponential form.

Solve $\log_4 x = 3$ . Convert: $x = 4^3 = 64$ .

Solve $\log_x 49 = 2$ . Convert: $x^2 = 49$ , so $x = 7$ (a logarithm base must be positive, so reject $-7$ ).

The Change-of-Base Formula

If you must compute a log your calculator does not have a button for, switch to a base it does have (10 or $e$ ):

$\log_b x = \frac{\log x}{\log b}$

So $\log_2 10 = \frac{\log 10}{\log 2} \approx \frac{1}{0.301} \approx 3.32$ .

A Reliable Game Plan

Evaluate a log? Ask "base to what power gives the argument?" Rewrite the argument as a power of the base.
See a product, quotient, or power inside a log? Use the sum, difference, or coefficient rule.
Solve a log equation? Convert to exponential form $b^y = x$ .
Weird base? Use change of base to base 10 or base $e$ .

Common Mistakes to Avoid

Splitting $\log_b(x + y)$ into $\log_b x + \log_b y$ — the product rule applies to multiplication, not addition.
Confusing the base with the argument when converting forms.
Forgetting that a log base and argument must be positive, so $\log_2(-8)$ is undefined.
Bringing the exponent down from the base instead of the argument.

Rewrite $\log_3 81 = 4$ in exponential form.

Evaluate $\log_5 125$ .

Solve $\log_2 x = 5$ for $x$ .

Write $\log_6 4 + \log_6 9$ as a single logarithm and evaluate it.

If $\log_b 2 = 0.43$ , what is $\log_b 8$ ?

Solve $\log_x 64 = 3$ for $x$ , where $x > 0$ .

Splitting the log of a sum: writing $\log_b(x + y) = \log_b x + \log_b y$ .

The sum rule applies only to a product inside the log: $\log_b(xy) = \log_b x + \log_b y$ . There is no rule for the log of a sum.

Swapping the base and the argument when converting, turning $\log_2 8 = 3$ into $8^3 = 2$ .

The definition is $\log_b x = y \Rightarrow b^y = x$ : base to the log-value equals the argument. So $\log_2 8 = 3$ means $2^3 = 8$ .

Bringing an exponent down from the base, as in reading $\log_{b^2} x = 2\log_b x$ .

The power rule pulls an exponent off the argument: $\log_b(x^k) = k\log_b x$ . An exponent on the base does not simply factor out the same way.

Accepting a negative or zero argument, e.g. evaluating $\log_3(-9)$ .

A logarithm is defined only for a positive argument (and a positive base $\ne 1$ ). No real exponent makes a positive base negative, so $\log_3(-9)$ is undefined.

What is the value of $\log_2 16$ ?

Which equation is equivalent to $\log_5 x = 3$ ?

What is the value of $\log_3 27 + \log_3 9$ ?

If $\log_4 x = \tfrac{3}{2}$ , what is the value of $x$ ?

If $\log_b 3 = k$ , which expression represents $\log_b 27$ ?

For what positive value of $x$ does $\log_x 8 = \tfrac{3}{2}$ ?