Unlock hundreds more features
Save your Quiz to the Dashboard
View and Export Results
Use AI to Create Quizzes and Analyse Results

OR
Sign inSign in with Facebook
Sign inSign in with Google
Quizzes > High School Quizzes > Mathematics

Exponential and Logarithmic Functions Practice Test

Sharpen your skills with engaging post test review

Difficulty: Moderate
Grade: Grade 10
Study OutcomesCheat Sheet
Paper art representing a trivia quiz on exponential and logarithmic concepts for high school algebra students.

What is the value of 2^3?
6
8
9
12
2 raised to the power of 3 means multiplying 2 by itself three times (2 x 2 x 2), which equals 8. This basic exponentiation concept is critical for understanding exponential functions.
What is the value of log₃(27)?
2
3
9
27
Since 3 raised to the power of 3 equals 27, log₃(27) is 3. Recognizing this inverse relationship between exponentiation and logarithms is key.
Which of the following correctly represents the relationship between exponential and logarithmic forms?
If log_b(a) = c, then b^c = a
If log_b(a) = c, then a^b = c
If log_b(a) = c, then c^b = a
If log_b(a) = c, then b^a = c
The definition of logarithms shows that if log_b(a) equals c, then the equivalent exponential form is b^c = a. This relationship is fundamental in converting between logarithmic and exponential expressions.
What is the domain of the function f(x) = log(x)?
x > 0
x ≥ 0
All real numbers
x < 0
The logarithmic function is defined only for positive arguments, so the domain is x > 0. Understanding domain restrictions is an important aspect of working with logarithms.
What transformation does the function f(x) = e^(x-2) represent compared to f(x) = e^x?
A horizontal shift 2 units to the right
A vertical shift 2 units upward
A reflection over the y-axis
A compression by a factor of 2
The expression e^(x-2) indicates a horizontal shift of the graph of e^x by 2 units to the right. Recognizing graph transformations is essential for understanding the behavior of exponential functions.
Solve for x: 2^(x+1) = 16.
2
3
4
5
Express 16 as 2^4 so that the equation becomes 2^(x+1) = 2^4. Equating the exponents gives x + 1 = 4, which leads to x = 3. This demonstrates the technique of solving exponential equations by equating exponents.
Solve for x: e^(2x) = 7.
x = ln(7)/2
x = 2ln(7)
x = ln(14)
x = 7/2
Taking the natural logarithm of both sides gives 2x = ln(7), so solving for x gives x = ln(7)/2. This question illustrates how logarithms can be used to solve exponential equations.
Solve for x: log₂(x) + log₂(x-2) = 3.
x = 4
x = 2
x = 6
x = 8
Combine the logarithms using the product rule to get log₂(x(x-2)) = 3, which implies x(x-2) = 2^3 = 8. Solving the resulting quadratic and considering domain restrictions leads to x = 4.
Simplify the expression: e^(ln 7).
7
ln 7
e^7
1
Since the exponential function and the natural logarithm are inverses, e^(ln 7) simplifies directly to 7. This property is a fundamental aspect of exponentiation and logarithms.
Which logarithmic law states that log_b(MN) = log_b(M) + log_b(N)?
Product Rule
Power Rule
Quotient Rule
Change-of-Base Rule
The product rule for logarithms indicates that the logarithm of a product equals the sum of the logarithms of the factors. This rule is essential for simplifying logarithmic expressions.
Solve for x: 3^(2x) = 81.
2
3
4
1
Since 81 can be written as 3^4, equate the exponents: 2x = 4, so x = 2. This exercise reinforces the method of writing expressions with a common base.
Express log₈(16) in terms of base 2 logarithms.
4/3
3/4
2
8/16
Using the change-of-base formula, log₈(16) = log₂(16)/log₂(8) = 4/3, because log₂(16) = 4 and log₂(8) = 3. This problem tests your ability to manipulate logarithms using different bases.
Solve the equation ln(x-1) = 2.
x = e² + 1
x = e² - 1
x = 2e
x = ln(2) + 1
Exponentiating both sides gives x - 1 = e², so adding 1 to both sides results in x = e² + 1. This exercise illustrates how to convert a logarithmic equation into its exponential form.
For the function f(x) = 2^x, what is its inverse function?
log₂(x)
2^(-x)
ln(x)/ln(2)
log₝₀(x)
The inverse of an exponential function is a logarithmic function with the same base, so the inverse of f(x) = 2^x is f❻¹(x) = log₂(x). This concept is crucial when working with inverse functions.
Solve the equation: 5^(x-1) = 125.
4
3
5
6
Since 125 is equal to 5^3, setting the exponents equal gives x - 1 = 3, so x = 4. This straightforward problem reinforces the method of solving exponential equations by expressing both sides with a common base.
Solve for x: 2^(x+2) = 3^(x-1).
x = (ln3 + 2ln2)/(ln3 - ln2)
x = (2ln3 + ln2)/(ln3 - ln2)
x = (2ln2 - ln3)/(ln2 - ln3)
x = (ln3 - 2ln2)/(ln3 + ln2)
Taking the natural logarithm of both sides yields (x + 2)ln2 = (x - 1)ln3. Rearranging and solving for x gives x = (ln3 + 2ln2)/(ln3 - ln2). This problem highlights the technique of using logarithms to solve equations with different bases.
Simplify the expression: log₂((16x²)/(8y)).
1 + 2log₂(x) - log₂(y)
2 + 2log₂(x) - log₂(y)
1 + log₂(x) - 2log₂(y)
2 + log₂(x²) - log₂(y)
First, simplify (16/8) to 2 so that the expression becomes log₂(2x²/y). Using logarithmic properties, this breaks down to log₂(2) + log₂(x²) - log₂(y), which simplifies to 1 + 2log₂(x) - log₂(y). This problem requires applying multiple logarithmic rules.
Find the inverse function of f(x) = 3^x - 4.
f❻¹(x) = log₃(x + 4)
f❻¹(x) = log₃(x) - 4
f❻¹(x) = 3^(x + 4)
f❻¹(x) = log₃(x - 4)
To find the inverse, set y = 3^x - 4 and solve for x: 3^x = y + 4, so x = log₃(y + 4). Replacing y with x, the inverse function is f❻¹(x) = log₃(x + 4). This tests your understanding of inverting exponential functions.
Solve the logarithmic equation: log₃(x + 5) + log₃(x - 1) = 1.
x = -2 + 2√3
x = -2 - 2√3
x = 2√3 - 1
x = 2 + 2√3
Combine the logarithms using the product rule: log₃[(x + 5)(x - 1)] = 1, meaning (x + 5)(x - 1) = 3. Expanding and solving the quadratic yields two solutions, but considering domain restrictions, the valid answer is x = -2 + 2√3.
Solve for x: 4^(x+1) = 8^(2x-3).
x = 11/4
x = 3
x = 9/4
x = 4
Rewrite 4 as 2² and 8 as 2³ so that the equation becomes 2^(2x+2) = 2^(6x-9). Equating the exponents gives 2x + 2 = 6x - 9, which simplifies to x = 11/4. This problem tests skills in manipulating exponents and solving equations.
0
{"name":"What is the value of 2^3?", "url":"https://www.quiz-maker.com/QPREVIEW","txt":"What is the value of 2^3?, What is the value of log₃(27)?, Which of the following correctly represents the relationship between exponential and logarithmic forms?","img":"https://www.quiz-maker.com/3012/images/ogquiz.png"}

Study Outcomes

  1. Analyze the properties of exponential functions, including growth and decay patterns.
  2. Apply logarithmic rules to solve various equations and simplify expressions.
  3. Demonstrate the conversion between exponential and logarithmic forms.
  4. Evaluate complex exponential and logarithmic expressions using key algebraic techniques.

Exponential & Logarithmic Post Test Cheat Sheet

  1. Understand Exponential Functions - Exponential functions of the form f(x)=bx (with b>0 and b≠1) model rapid growth or decay in everything from population booms to bank interest. Get comfortable with how changing the base affects the curve's steepness and direction. Mastering these basics turns abstract symbols into real-world insights. OpenStax: Exponential & Logarithmic Functions
  2. Graph Exponential Functions - Plotting these curves helps you see key features like the y‑intercept at (0,1), horizontal asymptotes, and how the graph shoots up or fades out. You'll learn to sketch behavior as x→∞ or x→−∞ in a snap. This visualization makes problem‑solving feel like reading a story. OpenStax: Introduction to Exponential & Logarithmic Functions
  3. Recognize the Natural Exponential Function - The function ex, where e≈2.718, is the superstar of continuous growth and decay models, from compound interest to radioactive clocks. Its unique rate-of-change property means the derivative of ex is itself. Getting comfy with 'e' unlocks a host of calculus shortcuts. OpenStax: Exponential & Logarithmic Functions
  4. Understand Logarithmic Functions - Logarithms are the inverse of exponentials: logb(x) answers "to what power must b be raised to get x?" They're essential for unraveling equations where the variable hides in an exponent. Knowing how these inverses work lets you switch back and forth with confidence. OpenStax: Exponential & Logarithmic Functions
  5. Convert Between Exponential and Log Forms - Turning by=x into logb(x)=y (and vice versa) becomes second nature with practice. This fluency is key to solving tricky equations and deciphering growth or decay in new contexts. Think of it as learning a secret code - once you crack it, everything clicks. OpenStax: Introduction to Exponential & Logarithmic Functions
  6. Master Logarithm Properties - Product Rule: logb(xy)=logb(x)+logb(y); Quotient Rule: logb(x/y)=logb(x)−logb(y); Power Rule: logb(xr)=r·logb(x). Using these shortcuts, you'll simplify monstrous expressions into neat sums and differences in no time. They're your toolkit for conquering complex log problems. OpenStax: Introduction to Exponential & Logarithmic Functions
  7. Solve Exponential Equations - Apply logarithms to both sides to "bring down" exponents and isolate the variable. For example, solving 2x=8 means taking log base 2 to find x=3. This technique turns intimidating exponentials into routine algebra. OpenStax: Exponential & Logarithmic Functions
  8. Use the Change-of-Base Formula - logb(x)=logc(x)/logc(b) lets you evaluate any log with just common (base 10) or natural (base e) logs on your calculator. No more struggling with odd bases - this formula is your universal adapter. OpenStax: Exponential & Logarithmic Functions
  9. Explore Real-World Applications - Exponentials and logs power everything from population forecasts and radioactive decay to the Richter scale for earthquakes and pH levels in chemistry. Seeing these functions in action gives you intuitive insight and makes abstract math feel downright practical. OpenStax: Introduction to Exponential & Logarithmic Functions
  10. Solve Logarithmic Equations - Combine or expand logarithms using their properties, then exponentiate both sides to find solutions. For example, log2(x)=3 rewrites as 23=x, so x=8. This two‑step process makes log equations clear and conquerable. OpenStax: Exponential & Logarithmic Functions
Make a Quiz (FREE) Copy & Edit This Quiz Create a Quiz with AI

Mathematics Quizzes

Powered by: Quiz Maker