📚 Introduction 📐 Key Properties ✅ Examples 📝 Practice 💡 ACT Tips

🔑 Core Exponential Properties

If $$y = b^x$$, then $$x = \log_b(y)$$

Example: Since $$2^3 = 8$$, we know that $$\log_2(8) = 3$$

Example 1: Solving an Exponential Equation

Question: Solve for $$x$$: $$3^{x+1} = 27$$

⏱️ ACT Time Tip: Always look for common bases first! Recognizing that 27 = 3³ makes this a 30-second problem.

Example 2: Solving a Logarithmic Equation

Question: Solve for $$x$$: $$\log_2(x) = 5$$

⏱️ ACT Time Tip: Converting logarithmic form to exponential form is the fastest way to solve. This takes 20-30 seconds!

Example 3: Using Logarithm Properties

Question: Simplify: $$\log_3(9) + \log_3(27)$$

⏱️ ACT Time Tip: Method 2 is faster when you recognize the powers immediately. Both methods work—choose the one you see first!

Example 4: Exponential Growth Application

Question: A population of bacteria doubles every 3 hours. If there are initially 500 bacteria, how many will there be after 9 hours?

⏱️ ACT Time Tip: For doubling/halving problems, count the periods and use powers of 2. Much faster than complex formulas!

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Practice Question 1

Solve for $$x$$: $$5^{2x} = 125$$

📖 Show Solution

Correct Answer: C) $$x = \frac{3}{2}$$

Solution:

Step 1: Express 125 as a power of 5:

$$125 = 5^3$$

Step 2: Rewrite the equation:

$$5^{2x} = 5^3$$

Step 3: Set exponents equal:

$$2x = 3$$

Step 4: Solve for x:

$$x = \frac{3}{2}$$

💡 Quick Tip: Memorize common powers: $$5^3 = 125$$, $$2^{10} = 1024$$, $$3^4 = 81$$

Practice Question 2

What is the value of $$\log_4(64)$$?

📖 Show Solution

Correct Answer: B) 3

Solution:

$$\log_4(64)$$ asks: “What power of 4 gives 64?”

$$4^1 = 4$$
$$4^2 = 16$$
$$4^3 = 64$$ ✓

Therefore, $$\log_4(64) = 3$$

Alternative method:

Convert to exponential form: $$4^x = 64$$
Express both as powers of 2: $$(2^2)^x = 2^6$$
$$2^{2x} = 2^6$$
$$2x = 6$$
$$x = 3$$

💡 Quick Tip: For simple logarithms, mentally test small powers. It’s faster than formal methods!

Practice Question 3

Simplify: $$\log_5(25) – \log_5(5)$$

📖 Show Solution

Correct Answer: B) 1

Solution Method 1: Using Quotient Property

$$\log_5(25) – \log_5(5) = \log_5\left(\frac{25}{5}\right)$$
$$= \log_5(5)$$
$$= 1$$ (because $$5^1 = 5$$)

Solution Method 2: Evaluate Each Term

$$\log_5(25) = \log_5(5^2) = 2$$
$$\log_5(5) = 1$$
$$2 – 1 = 1$$

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💡 Key Property: $$\log_b(b) = 1$$ for any base b. This appears frequently on the ACT!

Practice Question 4

If $$2^x = 16$$ and $$2^y = 8$$, what is the value of $$x + y$$?

📖 Show Solution

Correct Answer: C) 7

Solution:

Step 1: Solve for x:

$$2^x = 16 = 2^4$$
Therefore, $$x = 4$$

Step 2: Solve for y:

$$2^y = 8 = 2^3$$
Therefore, $$y = 3$$

Step 3: Find x + y:

$$x + y = 4 + 3 = 7$$

💡 Power of 2 Memorization: $$2^1=2, 2^2=4, 2^3=8, 2^4=16, 2^5=32, 2^6=64, 2^7=128, 2^8=256, 2^9=512, 2^{10}=1024$$

Practice Question 5

Which expression is equivalent to $$\log(x^3y^2)$$?

📖 Show Solution

Correct Answer: A) $$3\log(x) + 2\log(y)$$

Solution:

Step 1: Apply the product property:

$$\log(x^3y^2) = \log(x^3) + \log(y^2)$$

Step 2: Apply the power property to each term:

$$= 3\log(x) + 2\log(y)$$

Why other answers are wrong:

• B: Logarithms add, they don’t multiply
• C: Can’t combine the exponents like this
• D: The coefficients become exponents, not multipliers inside the log
• E: Incorrect application of properties

💡 Remember: Product → Add logs, Quotient → Subtract logs, Power → Multiply outside

💡 ACT Pro Tips & Tricks

Time Allocation

Allocate 45-75 seconds per exponential/logarithmic question. Simple evaluation problems (like $$\log_2(8)$$) should take 20-30 seconds. Equation-solving problems may need 60-75 seconds. Application problems (growth/decay) typically need the full 75 seconds.

When to Skip and Return

If you don’t immediately see how to express both sides with the same base, and the numbers don’t look familiar, mark it and move on. These questions often become clearer on a second pass. Don’t spend more than 90 seconds on any single exponential/log question.

Strategic Guessing

For logarithm evaluation questions, test the answer choices by converting to exponential form. For example, if asked for $$\log_3(81)$$, test: “Does 3⁴ = 81?” This verification method is often faster than solving directly and helps eliminate wrong answers quickly.

Quick Verification Method

Always verify exponential solutions by substituting back. If you found $$x = 3$$ for $$2^x = 8$$, check: $$2^3 = 8$$ ✓. This 5-second check catches calculation errors and gives you confidence. For logarithms, convert your answer to exponential form to verify.

Common Trap Answers

Calculator Usage

Use your calculator for verification, not primary solving. The LOG button gives base-10 logarithms, LN gives natural logarithms (base e). For other bases, use change of base: $$\log_5(20) = \frac{\log(20)}{\log(5)}$$. But remember: mental math with common powers is usually faster!

Question Type Recognition

Mistake #1: Adding Exponents When Multiplying Different Bases

The Error: Thinking $$2^3 \cdot 3^2 = 6^5$$ or similar incorrect combinations.
The Fix: You can only add exponents when the bases are the same: $$2^3 \cdot 2^2 = 2^5$$. Different bases must be calculated separately: $$2^3 \cdot 3^2 = 8 \cdot 9 = 72$$.

Mistake #2: Confusing Product and Sum in Logarithms

The Error: Writing $$\log(x + y) = \log(x) + \log(y)$$.
The Fix: $$\log(x) + \log(y) = \log(xy)$$ (product, not sum). There’s no simple formula for $$\log(x + y)$$—it stays as is!

Mistake #3: Forgetting to Check Domain Restrictions

The Error: Solving $$\log(x – 3) = 2$$ and getting $$x = 103$$, but not checking if it’s valid.
The Fix: Logarithms require positive arguments. After solving, verify that $$x – 3 > 0$$. In this case, $$103 – 3 = 100 > 0$$, so it’s valid. Always check domain restrictions!

Mistake #4: Mishandling Negative Exponents

The Error: Thinking $$2^{-3} = -8$$ or $$2^{-3} = -\frac{1}{8}$$.
The Fix: Negative exponents mean reciprocal: $$2^{-3} = \frac{1}{2^3} = \frac{1}{8}$$. The result is positive! The negative sign affects position (numerator vs. denominator), not the sign of the answer.

Finance & Economics

Compound interest follows exponential growth: $$A = P(1 + r)^t$$. If you invest $1,000 at 5% annual interest, after 10 years you’ll have $$1000(1.05)^{10} \approx \$1,629$$. Understanding exponential functions helps you make smart financial decisions about savings, investments, and loans. Credit card debt also grows exponentially—which is why minimum payments can trap people in debt cycles.

Science & Medicine

Radioactive decay follows exponential patterns. Carbon-14 dating uses the equation $$N(t) = N_0 \cdot e^{-kt}$$ to determine the age of ancient artifacts. In medicine, drug concentration in the bloodstream decreases exponentially over time. Doctors use logarithmic scales (like pH for acidity or decibels for sound) because they compress large ranges into manageable numbers—pH 7 to pH 14 represents a 10-million-fold difference in hydrogen ion concentration!

Technology & Data Science

Computer scientists use logarithms constantly. Binary search algorithms run in $$O(\log n)$$ time—searching through 1 million items takes only about 20 steps! Data compression, cryptography, and machine learning all rely heavily on exponential and logarithmic functions. Moore’s Law (computing power doubles every 18-24 months) is exponential growth in action.

Population & Epidemiology

Population growth (and disease spread) often follows exponential patterns initially. The COVID-19 pandemic demonstrated exponential growth dramatically—when each infected person infects 2-3 others, cases double rapidly. Understanding these patterns helps public health officials make critical decisions about interventions. Conversely, population decline and species extinction also follow exponential decay models.

Q1: What’s the easiest way to solve exponential equations on the ACT?

Answer: The fastest method is expressing both sides with the same base, then setting the exponents equal. For example, to solve $$4^x = 64$$, recognize that $$4 = 2^2$$ and $$64 = 2^6$$, so $$(2^2)^x = 2^6$$, which gives $$2^{2x} = 2^6$$, therefore $$2x = 6$$ and $$x = 3$$. This method takes 20-30 seconds once you memorize common powers. If you can’t find a common base within 10 seconds, take the logarithm of both sides instead.

Q2: How do I remember which logarithm property to use?

Answer: Use this simple pattern: logarithms turn multiplication into addition, division into subtraction, and exponents into multiplication. Specifically: $$\log(ab) = \log(a) + \log(b)$$ (product becomes sum), $$\log(a/b) = \log(a) – \log(b)$$ (quotient becomes difference), and $$\log(a^n) = n\log(a)$$ (power comes out front). Think of logarithms as “breaking down” operations into simpler ones. Practice with this mnemonic: “Products Add, Quotients Subtract, Powers Multiply.”

Q3: When should I use my calculator for exponential and logarithmic problems?

Answer: Use your calculator primarily for verification and for non-standard bases. For example, to evaluate $$\log_7(50)$$, use the change of base formula: $$\frac{\log(50)}{\log(7)}$$ or $$\frac{\ln(50)}{\ln(7)}$$. However, for common problems like $$2^5$$ or $$\log_3(27)$$, mental math is faster. Your calculator is also helpful for word problems involving compound interest or exponential growth where the numbers aren’t “nice.” Always try mental math first—if you don’t see the answer in 10 seconds, reach for the calculator.

Q4: What’s the difference between log, ln, and log with a subscript?

Answer: $$\log(x)$$ typically means $$\log_{10}(x)$$ (common logarithm, base 10), $$\ln(x)$$ means $$\log_e(x)$$ (natural logarithm, base e ≈ 2.718), and $$\log_b(x)$$ means logarithm with base b. On the ACT, you’ll see all three. Your calculator has LOG and LN buttons for base 10 and base e. For other bases, use the change of base formula: $$\log_b(x) = \frac{\log(x)}{\log(b)} = \frac{\ln(x)}{\ln(b)}$$. The properties work the same way regardless of base!

Q5: How can I quickly recognize exponential growth vs. decay problems?

Answer: Look at the base in the exponential function $$f(x) = a \cdot b^x$$. If $$b > 1$$, it’s growth (the quantity increases). If $$0 < b < 1$$, it's decay (the quantity decreases). For example, $$f(x) = 100(2)^x$$ is growth (doubling), while $$f(x) = 100(0.5)^x$$ is decay (halving). In word problems, look for keywords: "doubles," "triples," "increases by" suggest growth; "halves," "decreases by," "decays" suggest decay. Also, growth curves go up to the right, decay curves go down to the right when graphed.

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