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Quizzes > High School Quizzes > Mathematics

Unit 4 Exponential & Logarithmic Practice Quiz

Ace Units 4 and 5 with Guided Solutions

Difficulty: Moderate
Grade: Grade 9
Study OutcomesCheat Sheet
Colorful paper art promoting the Unlock Exponential Mastery math quiz for high school students.

Easy
What is the general form of an exponential function?
y = ax^2
y = ax + b
y = ab^x
y = a log_b(x)
Exponential functions are characterized by a constant raised to a variable power. The expression y = ab^x correctly reflects this property, where a is the initial value and b is the base.
If f(x) = 3^x, what is f(2)?
6
12
3
9
Substituting x = 2 into the function f(x) = 3^x gives f(2) = 3^2, which equals 9. This demonstrates how to evaluate simple exponential expressions.
Which of the following graphs best represents an exponential growth function?
A graph that rises steeply to the right
A horizontal line
A parabola opening upward
A straight line with a constant slope
Exponential growth functions increase rapidly as x increases, which is best illustrated by a graph that rises steeply to the right. The other options describe linear, quadratic, or constant behaviors.
What is the horizontal asymptote of the exponential function y = ab^x (with a ≠0 and b > 0, b ≠1)?
y = a
y = 0
x = a
x = 0
For exponential functions of the form y = ab^x, as x becomes very negative, the function approaches 0, hence the horizontal asymptote is y = 0. This property distinguishes it from other types of functions.
What is the inverse function of f(x) = 2^x?
fā»Ā¹(x) = logā‚‚(x)
fā»Ā¹(x) = x²
fā»Ā¹(x) = ln(x)
fā»Ā¹(x) = 2/x
The inverse of an exponential function is a logarithmic function with the same base. Since f(x) = 2^x, its inverse is given by fā»Ā¹(x) = logā‚‚(x).
Medium
Solve the equation: 2^(x+1) = 16.
3
-3
4
2
Express 16 as a power of 2; 16 is 2^4. Therefore, 2^(x+1) = 2^4 implies that x + 1 = 4, so x = 3. This demonstrates the method of equating exponents when the bases are the same.
Determine the domain of the function f(x) = log(x - 3).
x > 3
x ≄ 3
All real numbers
x < 3
For logarithmic functions, the argument must be positive. Since the argument here is (x - 3), we require x - 3 > 0, hence x > 3. This is a key property of logarithms.
Which logarithm property allows you to combine logā‚(x) + logā‚(y) into a single logarithm?
logā‚(x/y)
logā‚(x + y)
logā‚(x - y)
logā‚(xy)
The product rule for logarithms states that logā‚(x) + logā‚(y) = logā‚(xy). This property is fundamental when simplifying or combining logarithmic expressions.
Solve for x: logā‚ƒ(x) = 4.
64
12
1/81
81
By rewriting the logarithmic equation in exponential form, we have x = 3ā“, which equals 81. This illustrates the conversion between logarithmic and exponential forms.
What is the inverse of the function f(x) = 5^x?
fā»Ā¹(x) = logā‚…(x)
fā»Ā¹(x) = logā‚ā‚€(x)
fā»Ā¹(x) = xāµ
fā»Ā¹(x) = 5/x
The inverse of an exponential function is found by taking the logarithm with the same base. Hence, the inverse of f(x) = 5^x is fā»Ā¹(x) = logā‚…(x).
What is the primary difference between exponential growth and exponential decay?
The exponent is negative for growth
The graph is a straight line for decay
The base is greater than 1 for growth and between 0 and 1 for decay
The exponent is always zero in decay
Exponential growth functions have a base greater than 1, resulting in increasing values as x increases. Conversely, exponential decay functions use a base between 0 and 1, causing the function to decrease.
How can the function f(x) = e^(2x) be rewritten using the properties of exponents?
f(x) = (e²)^x
f(x) = 2e^x
f(x) = e^(x²)
f(x) = x^(e²)
Using the exponent rule e^(2x) = (e²)^x simplifies the expression. This property of exponents allows us to view exponential functions in alternative forms.
Which of the following is equivalent to ln(1/e^x)?
ln(e^x)
1/x
-x
x
By using logarithm properties, ln(1/e^x) can be rewritten as ln(e^(-x)), which simplifies directly to -x. This demonstrates the power rule for logarithms in action.
Simplify the expression: logā‚‚(8) + logā‚‚(4).
6
7
5
2
Since logā‚‚(8) equals 3 and logā‚‚(4) equals 2, their sum is 3 + 2 = 5. This problem reinforces the evaluation of logarithms with known powers of the base.
Consider the function f(x) = 3^(x - 2). What is f(2)?
3
9
0
1
Substituting x = 2 into the function yields 3^(2 - 2) = 3^0, which is equal to 1. This demonstrates the critical principle that any non-zero number raised to the zero power is 1.
Hard
Solve the equation: 4^(x + 1) = 32.
2
4
5
3/2
Rewrite 4 as 2² so that 4^(x + 1) becomes (2²)^(x + 1) = 2^(2x + 2). Since 32 is 2^5, equate the exponents: 2x + 2 = 5, which gives x = 3/2. This method relies on expressing both sides with a common base.
If f(x) = 2^(3x) and g(x) = 8^x, are these functions equivalent?
No, they are not equivalent
Only for x positive
Only for x negative
Yes, they are equivalent
Recognize that 8 can be written as 2³. Thus, g(x) = 8^x becomes (2³)^x = 2^(3x), which is exactly f(x). This demonstrates the importance of rewriting expressions to reveal underlying equivalences.
Solve for x: 3^(2x) = 27.
3/2
9
1
27/2
Express 27 as 3³. Setting 3^(2x) equal to 3³ allows you to equate the exponents: 2x = 3, hence x = 3/2. This problem reinforces solving exponential equations by comparing powers.
Express the logarithmic equation log_b(x²) = 2 log_b(x) by identifying the logarithm property used.
Change-of-Base Formula
Quotient Rule
Power Rule
Product Rule
The power rule of logarithms states that log_b(x^c) equals c log_b(x). Here, log_b(x²) becomes 2 log_b(x), which is a direct application of the power rule.
Given the function f(x) = logā‚‚(x), if f(8) = 3, what is the value of f(16)?
3
5
4
16
Since logā‚‚(8) = 3 and 8 = 2³, for 16 = 2ā“ the logarithm f(16) is logā‚‚(16) which equals 4. This problem shows how logarithms translate exponents into numerical values.
0
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Study Outcomes

  1. Understand the fundamental properties and behavior of exponential functions.
  2. Analyze and solve exponential equations using appropriate methods.
  3. Apply logarithmic transformations to simplify and solve problems involving exponentials.
  4. Graph exponential functions and interpret their key characteristics.
  5. Evaluate real-world situations modeled by exponential growth and decay.

Unit 4 Exponential & Log Answer Key Cheat Sheet

  1. Understand the definition of exponential functions - Exponential functions take the form f(x) = a Ā· bx, where "a" is your starting amount and "b" is the rate at which things grow or shrink. Once you spot how changing "b" tweaks the curve, you'll feel like a math magician! Learn more on GeeksforGeeks
  2. Master the properties of exponents - Rules like ax Ā· ay = ax+y and (ax)y = axy let you simplify beasts of expressions into cute little kittens. Practice these tricks until they're second nature - your future self will thank you when exams roll around! Check out rules on GeeksforGeeks
  3. Differentiate growth vs. decay - If the base b is bigger than 1, you've got exponential growth; if it's between 0 and 1, that's graceful decay. Think population booms versus radioactive half-lives - both use the same formula, but the story changes with "b"! Explore examples on GeeksforGeeks
  4. Use the exponential growth formula - With y = a(1 + r)x, "r" is your growth rate (like interest or population increase). Plug in real numbers and watch your graphs climb faster than your phone battery drains at a party! Dive into details on GeeksforGeeks
  5. Apply the exponential decay formula - The mirror image is y = a(1 - r)x, where "r" is how fast things fade away - think half-life in radioactive materials or depreciation on gadgets. Recognizing decay will help you predict when that old smartphone is truly "vintage." Read more on GeeksforGeeks
  6. Practice solving exponential equations - Turn both sides into the same base and match exponents to crack the code. This method feels like unlocking a secret level in your favorite video game - super satisfying when you get it! Solve examples on BYJU'S
  7. Visualize the exponential graph - Check out the classic J‑shaped curve (or reverse J for decay), spot the horizontal asymptote, and note where it crosses the y‑axis at a. The more you sketch, the more intuitive these soaring (or sinking) curves become. See graph guides on GeeksforGeeks
  8. Learn exponential derivatives - You get d/dx (ex) = ex and d/dx (ax) = axĀ·ln(a). These formulas are golden stamps when you tackle calculus problems - no sweat required once you memorize them! Review on GeeksforGeeks
  9. Understand exponential integrals - The antiderivatives are just as neat: ∫ex dx = ex + C and ∫ax dx = ax/ln(a) + C. This means you can reverse-engineer area and accumulation problems with ease - math magic unlocked! Explore more on GeeksforGeeks
  10. Apply to real-world scenarios - Use exponentials to model population booms, radioactive decay, compound interest, and even virus spread. Seeing math animate real life adds an extra thrill to every calculation. Check applications on GeeksforGeeks
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