Logarithms: Exponents, Turned Inside Out
Understand logarithms as the inverse of exponents, master the product, quotient, and power rules, and solve log and exponential equations — for the ACT, with worked problems.
The Short Version
- logb(x) = y means b₋ = x — a log asks "what exponent gives x?"
- A logarithm is the inverse of an exponential function.
- Rules: product → add logs, quotient → subtract, power → move the exponent in front.
- Solve exponential equations by taking a log; solve log equations by rewriting in exponential form. An ACT and Algebra II topic.
Exponents take a base and a power and give you a result: 2³ = 8. A logarithm runs that backward — it starts with the result and asks for the power: "2 to what equals 8?" The answer, 3, is log2(8). That single reframing — a log is just the missing exponent — demystifies the whole topic, including the rules that look intimidating at first and the equations that seem unsolvable.
This guide builds logs from their definition, lays out the three rules, and shows how to solve log and exponential equations, with worked and practice problems matched to real test difficulty at Northside Tutoring.
Why Logarithms Matter
Logarithms appear on the ACT and are central to Algebra II and Pre-Calculus. They model anything that grows or shrinks multiplicatively — sound (decibels), earthquakes (Richter scale), pH, and exponential growth. They're beyond the SSAT and tested more on the ACT than the SAT.
What a Logarithm Is
The statement logb(x) = y is just another way of writing b₋ = x. The base b is the same in both; the log gives you the exponent y. So log2(8) = 3 because 2³ = 8, and log10(100) = 2 because 10² = 100.
Logs Undo Exponents
Because a log is the inverse of an exponential, their graphs are mirror images across the line y = x:
y = log₂ x is the reflection of y = 2ˣ across the line y = x — they undo each other.
The Three Log Rules
The rules mirror the exponent rules — which makes sense, since logs are exponents:
| Rule | Form |
|---|---|
| Product | log(mn) = log m + log n |
| Quotient | log(m/n) = log m − log n |
| Power | log(mₖ) = k · log m |
Change of Base
To compute a log in any base on a calculator (which only has log base 10 and natural log), use the change-of-base formula:
Solving Log & Exponential Equations
Two mirror-image techniques:
- Exponential equation (variable in the exponent): rewrite as a log or take a log of both sides. 2ˣ = 16 → x = log2(16) = 4.
- Log equation (variable inside a log): rewrite in exponential form. log3(x) = 2 → x = 3² = 9.
Check the domain
You can only take the log of a positive number. After solving a log equation, discard any solution that would make you take the log of zero or a negative — an extraneous solution.
Where You'll See This — Test by Test
No reference sheet covers log rules. The ACT tests logarithms — their definition, rules, and equations — more than the SAT. They're beyond the SSAT.
ACT
Tests evaluating logs, applying the product/quotient/power rules, and solving log/exponential equations.
Explore ACT Tutoring → College AdmissionsSAT
The SAT rarely tests logarithms directly; this is more of an ACT and Algebra II topic.
Explore SAT Tutoring → K-12 CurriculumAlgebra II
A core Algebra II topic and the inverse of exponential functions.
Explore Algebra Tutoring → K-12 CurriculumPre-Calculus
Pre-Calculus develops logs further, including the natural log and applications.
Explore Math Tutoring →Watch the Lesson
Sometimes a diagram needs a voice. In the short video below, one of our Northside tutors walks through the core idea and works through test-style problems in real time.
Logarithms — In Plain English
A live walkthrough from our tutoring team.
— Featuring a Northside Tutoring instructor
Worked Example Problems
These problems are calibrated to the difficulty you'll actually see on test day. Try each one before opening the solution.
Evaluate log₂(16).
Show solution
Ask: 2 to what power is 16? 2⁴ = 16, so the answer is 4.
Solve 3ˣ = 81.
Show solution
81 = 3⁴, so x = 4. (Or x = log381 = 4.)
Solve log₅(x) = 3.
Show solution
Rewrite in exponential form: x = 5³ = 125.
Expand log(xy³) using log rules.
Show solution
Product rule then power rule: log x + log y³ = log x + 3 log y.
Write as a single log: log 6 − log 2.
Show solution
Quotient rule: log(6/2) = log 3.
Common Mistakes to Avoid
Three traps that catch students every year
- Confusing the base and the argument. logb(x) asks for the exponent on b that gives x — keep the base straight.
- Splitting a log of a sum. log(m + n) does not equal log m + log n. The product rule applies to multiplication, not addition.
- Keeping extraneous solutions. You can't take the log of zero or a negative — discard any solution that does.
Practice Problems — You Try
Three problems below. Work each before checking the solution.
Evaluate log₁₀(1000).
Show solution
10³ = 1000, so the answer is 3.
Solve 2ˣ = 32.
Show solution
32 = 2⁵, so x = 5.
Solve log₂(x) + log₂(x − 2) = 3.
Show solution
Combine: log2[x(x−2)] = 3 → x(x−2) = 2³ = 8 → x² − 2x − 8 = 0 → (x−4)(x+2)=0. x = 4 or −2; discard −2 (log of a negative), so x = 4.
The Northside Method — How We Teach This 1-on-1
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- Assessment. We diagnose which specific skills are slowing your student down — not just whether they "get it" in the abstract.
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