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Quizzes > Mathematics > Algebra

Exponential & Logarithmic Functions Quiz - Can You Ace It?

Ready for a post-lesson exponential and logarithmic functions quiz? Start your practice test now!

Difficulty: Moderate
2-5mins
Learning OutcomesCheat Sheet
Paper art style quiz illustration showing exponential and logarithmic graphs and equations on golden yellow background

This post test helps you practice exponential and logarithmic functions and check gaps before an exam. You get instant scoring and feedback as you solve growth, decay, and equation problems, so you can review Algebra II skills and see progress. For extra log practice, try the log quiz .

What is the value of 2^3 * 2^4?
128
2
64
4096
When multiplying expressions with the same base, you add the exponents: 3 + 4 = 7, so 2^7 = 128. This property of exponents simplifies many computations. For more properties of exponents, see .
Solve for x: 5^x = 125.
x = 125
5
3
25
Since 125 = 5^3, you set x = 3 to satisfy 5^x = 5^3. This uses the one-to-one property of exponential functions. See more at .
What is log10(1000)?
0
1000
3
1
log10(1000) asks "10 to what power gives 1000?" Since 10^3 = 1000, the log base 10 of 1000 is 3. Visit for details.
Evaluate ln(e^5).
1
0
e^5
5
The natural logarithm ln and the exponential function e^x are inverses, so ln(e^5) = 5. This is a standard property of logarithms and exponentials. More information at .
What is the value of 3^0?
1
3
0
-1
Any nonzero number to the zero power equals 1 by definition of exponents. This arises from the property a^m / a^m = a^{m?m} = a^0 = 1. See for more.
Convert the statement log2(8) = 3 into its exponential form.
2^8 = 3
2^3 = 8
3^2 = 8
8^2 = 3
By definition of logarithms, log_b(a) = c is equivalent to b^c = a. Here b=2, a=8, c=3 so 2^3 = 8. More discussion at .
What is log5(1)?
5
-1
1
0
For any base b>0, log_b(1) = 0 since b^0 = 1. This is a fundamental property of logarithms. See for reference.
Simplify (10^2)^3.
10^5
10^6
10^8
10^1
When raising a power to another power, you multiply exponents: (10^2)^3 = 10^{2*3} = 10^6. This is one of the core exponent rules. More details at .
Solve for x: 4^(2x - 1) = 16.
x = 0
x = 1
x = -1
x = 2
Rewrite 16 as 4^2, so 4^(2x - 1) = 4^2 implies 2x - 1 = 2. Solving gives x = 1. This uses the one-to-one property of exponential functions. See .
Find x if ln(x) = 3.
x = ln(3)
x = 3 ln(e)
x = e^3
x = 3e
To solve ln(x) = 3, exponentiate both sides: x = e^3. This reverses the natural logarithm. More at .
Using change of base, evaluate log_3(81).
3
9
1/4
4
81 = 3^4, so log_3(81) = 4 directly. Change-of-base isn't needed here but could be used: log(81)/log(3) = 4. For change-of-base details see .
Solve for x: log2(x + 1) = 4.
x = 16
x = 7
x = 8
x = 15
Rewrite in exponential form: x + 1 = 2^4 = 16, so x = 15. This uses the definition of logarithms. Learn more at .
What is the value of e^{ln(7)}?
1/7
7
e^7
ln(7)
e^{ln(7)} = 7 because exponentials and natural logs are inverse functions. For the inverse property, see .
Solve for x: 3^{x + 2} = 2 * 3^x.
x = log3(2) - 2
x = log3(2)
x = -2 + 1
x = -2 + log3(2)
Divide both sides by 3^x: 3^2 = 2, so 9 = 2, which is incorrect. Actually rewrite 3^{x+2}/3^x = 3^2 = 9, so 9 = 2 gives no solution, correct process is wrong. Actually set 3^{x+2} = 3^x * 2 ? 3^2 = 2 ? contradiction. There is no real solution. For solving exponentials see .
Simplify log_a(a^5).
5
a^5
log_a(5)
1/5
log base a of a^5 equals 5 because logarithms undo exponentiation. This is the basic log identity log_b(b^x)=x. See .
Find x if log10(x) = -1.
10
-1
1
0.1
Rewrite in exponential form: x = 10^{-1} = 0.1. This uses the definition of common logarithms. More info at .
Solve for x: 2^x = x^2.
x = 2 or x = 4
x = 0 or x = 1
x = -2 or x = 4
x = 1 or x = 2
Test integer solutions: for x=2, 2^2=4 and 2^2=4; for x=4, 2^4=16 and 4^2=16. No other real solutions exist. See analysis at .
Solve ln(x) + ln(x - 1) = 0 for x > 1.
x = ?2
x = 1
x = 2
x = (1 + ?5)/2
Combine logs: ln[x(x-1)] =0 implies x(x-1)=1. Solve x^2 - x -1=0 giving x = (1±?5)/2; only positive >1 is (1+?5)/2. More at .
What is the derivative of f(x) = e^{2x}?
2e^{2x}
e^{x}
2xe^{2x}
e^{2x}
The derivative of e^{2x} is e^{2x} times the inner derivative 2, giving 2e^{2x}. This uses chain rule. For more, see .
Solve for x: log2(x - 3) = 4.
x = 7
x = 19
x = 20
x = 16
Convert to exponential form: x - 3 = 2^4 = 16, so x = 19. Domain requires x - 3>0, which is satisfied. See .
Simplify log5(25) + log5(125).
5
3
4
6
log5(25)=2 and log5(125)=3, so their sum is 5. Oops, check: 25=5^2 and 125=5^3, sum=2+3=5. So answer is 5. .
Solve e^{3x} = 7 for x.
x = e^{7/3}
x = ln(7^3)
x = 3 ln(7)
x = ln(7)/3
Take natural log: 3x = ln(7), so x = ln(7)/3. This isolates x using inverse of the exponential. More at .
Evaluate the limit lim_{x->0} (e^x - 1)/x.
Infinity
1
0
e
Using the series expansion of e^x or L'Hôpital's Rule, the limit equals 1. This is a fundamental limit in calculus. See .
Solve for x in x * e^x = 5.
x = 5e
x = ln(5)
x = W(5)
x = e^5 /5
Equations of the form x e^x = k are solved using the Lambert W function: x = W(k). Here x = W(5). For more on the Lambert W, see Lambert W Function.
Evaluate the indefinite integral ? dx/(x ln x).
ln|x| + C
ln(x ln x) + C
ln|ln x| + C
1/ln x + C
Let u = ln x, then du = dx/x, so the integral becomes ? du/u = ln|u| + C = ln|ln x| + C. This substitution is standard. See .
0
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Study Outcomes

  1. Understand the inverse relationship between exponential and logarithmic functions -

    Recognize how logarithms reverse the process of exponentiation and identify corresponding pairs of exponential and logarithmic forms.

  2. Apply properties of exponents to simplify expressions -

    Use product, quotient, and power rules to streamline exponential terms and prepare them for equation solving.

  3. Solve exponential equations using algebraic techniques and logarithms -

    Isolate variables in the exponent and apply logarithmic operations to find exact solutions to exponential equations.

  4. Apply logarithmic properties to simplify and evaluate expressions -

    Leverage product, quotient, and change”of”base rules to rewrite complex logarithmic expressions in manageable forms.

  5. Convert between exponential and logarithmic forms -

    Rewrite equations interchangeably as exponential or logarithmic statements to reveal different perspectives on the same relationship.

  6. Analyze real”world scenarios involving exponential growth and decay -

    Interpret and model applications like population growth or radioactive decay using exponential and logarithmic functions.

Cheat Sheet

  1. Understanding Exponential Growth and Decay -

    Exponential functions take the form f(x)=a·b^x, where a≠0 and b>0. When b>1 you have growth and when 0

  2. Logarithms as the Inverse of Exponentials -

    A logarithmic function log_b(x)=y answers "b^y=x" and reverses the effect of an exponential. This inverse relationship lets you solve for exponents, and the key is remembering that log_b(b^x)=x and b^(log_b x)=x. Mnemonic tip: "logs undo exponents" so when you see b^x inside a log you can simplify instantly.

  3. Core Logarithm Laws -

    The product, quotient, and power rules (log_b(M·N)=log_b M+log_b N; log_b(M/N)=log_b M - log_b N; log_b(M^k)=k·log_b M) are essential for simplifying expressions. These laws make it easy to break down complex sums or differences of logs when tackling your exponential and logarithmic functions quiz. Practice combining logs with LIP (Log of a Product) and QLQ (Quotient Log Quick) as memory tricks.

  4. Change-of-Base Formula -

    When you need to compute logs in different bases, use log_b(x)=log_k(x)/log_k(b) for any convenient base k (often 10 or e). This formula is invaluable on calculators that only offer log base 10 or ln, turning tricky base-7 or base-2 logs into familiar territory. Armed with this, you'll breeze through any exponential functions assessment or logarithmic functions practice test.

  5. Graphing Exponentials and Logarithms -

    Key features include a horizontal asymptote (y=0 for exponentials, x=0 for logarithms), intercepts at (0,a) or (1,0), and distinctive curvature. Understanding how vertical/horizontal shifts and reflections affect these graphs gives you full control when analyzing transformations on a post test exponential and logarithmic functions quiz. Sketch a few f(x)=2^x+3 or g(x)=log_2(x - 1) to see shifts in action!

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