Logarithms on the ACT

A logarithm answers one question: what exponent do I need? That is the whole concept. Logs show up less often than lines, triangles, or percentages, but they are very learnable, and skipping them means choosing to miss a question you could have finished in seconds.

The definition, in the only form worth memorizing

The statement log base b of x equals y means the same thing as b to the power y equals x. The base stays the base, the log is the exponent, and the number inside the log is the result.

So log base 2 of 8 equals 3, because 2 to the third power is 8. Log base 5 of 25 equals 2, because 5 squared is 25. Log base 3 of 81 equals 4, because 3 to the fourth power is 81. Translate any log into an exponent sentence and the question usually answers itself.

Two special cases fall right out of that. Log base b of 1 is always 0, since anything to the zero power is 1. Log base b of b is always 1. And log base 2 of one eighth equals -3, because 2 to the power -3 is one eighth. Negative logs are legal, negative inputs are not.

Converting between the two forms

Most ACT log questions solve themselves the instant you switch forms, so practice going both directions.

• Log base 10 of 1000 equals 3 becomes 10 cubed equals 1000.
• 7 squared equals 49 becomes log base 7 of 49 equals 2.
• If log base 6 of x equals 2, convert to 6 squared equals x, so x = 36.
• If log base x of 27 equals 3, convert to x cubed equals 27, so x = 3.

When a log is written with no base, like log 100, the base is understood to be 10, so log 100 equals 2. The ACT gives you no formula sheet, so this conversion is worth memorizing. The broader list lives in the ACT math formulas guide.

The three rules

These come straight from the exponent rules you already know, since a log is an exponent.

• Product rule. Log of a product is the sum of the logs. Log base 2 of 4 plus log base 2 of 8 equals log base 2 of 32, which is 5. Check it slowly: the pieces are 2 and 3, and 2 plus 3 is 5.
• Quotient rule. Log of a quotient is the difference of the logs. Log base 3 of 54 minus log base 3 of 2 equals log base 3 of 27, which is 3.
• Power rule. An exponent inside the log comes out front. Log base 5 of 25 cubed equals 3 times log base 5 of 25, which is 3 times 2, or 6. That checks out, because 25 cubed is 15625, and 5 to the sixth power is also 15625.

These rules trace back to how exponents combine, so if that material is rusty, a pass through exponent rules makes logs feel much less foreign.

Solving exponential equations

The most common ACT use of logs is undoing an exponent. When both sides share a base, you do not even need the log.

• Solve 2 to the power x equals 32. Since 32 is 2 to the fifth, x = 5.
• Solve 3 to the power (x + 1) equals 81. Since 81 is 3 to the fourth, set x + 1 = 4, so x = 3. Verify: 3 to the fourth is 81.

When the variable is inside the log instead, convert to exponential form. Solve log base 2 of (x - 1) equals 4. That becomes 2 to the fourth equals x - 1, so 16 = x - 1 and x = 17. Check it: 17 minus 1 is 16, and log base 2 of 16 is 4.

Common traps

• Splitting the log of a sum. The log of a sum does not break apart. Log base 2 of (4 plus 4) is log base 2 of 8, which is 3, while log base 2 of 4 plus log base 2 of 4 is 2 plus 2, which is 4. Those are different numbers. The product rule applies to multiplication inside the log, never to addition.
• Flipping the base and the answer. Log base 2 of 8 is 3, not 8 divided by 2 and not log base 8 of 2. Say the exponent sentence out loud first.
• Accepting a negative or zero input. You cannot take the log of a negative number or of 0, so discard any solution that produces one.
• Misplacing the power rule. The exponent has to be on the entire inside expression before you bring it out front.

A short drill

Evaluate log base 3 of 9, log base 4 of 64, and log base 10 of 0.01. Then simplify log base 6 of 4 plus log base 6 of 9, and solve 5 to the power (2x) equals 125. The answers are 2, 3, -2, 2, and x = 1.5. That last one works because 125 is 5 cubed, so 2x = 3. Ten minutes on problems like these usually turns logs from an automatic miss into a reliable point, and thirty-six has concept quizzes that isolate them. Logs are one of several topics worth a deliberate visit rather than a hope, which is the theme of the hardest ACT math topics, and they sit alongside the rest of the material in the ACT Math functions hub.

This article offers general ACT prep guidance. The ACT can change from year to year, including its format, scoring, policies, test dates, and fees, so always confirm the latest details on the official ACT website at act.org before you make decisions. ACT® is a registered trademark of ACT, Inc. thirty-six is not affiliated with, endorsed by, or sponsored by ACT.