Exponential & Logarithmic Functions Practice Questions, Quiz, and Step-by-Step Lesson - improve your math skills with focused questions and clear explanations.
Exponential & Logarithmic Functions Practice Quiz with a Step-by-Step Interactive Lesson
Use the quiz at the top of the page to practice exponential and logarithmic functions with the most important skills for algebra and precalculus: exponential functions \(b^x\) and \(ab^x\), domain and range, horizontal asymptotes, and graph transformations, exponential growth and exponential decay, the inverse relationship between exponentials and logs, logarithms \(\log_b(x)\), including the common logarithm \(\log_{10}(x)\) and the natural logarithm \(\ln(x)\), core log rules (product, quotient, and power), the change of base formula, and the most common problem types: solve exponential equations and solve logarithmic equations (with correct domain checks). If you want a refresher with clear steps, click Start lesson to open a guided mini-book with worked examples and quick checks.
How this exponential and logarithmic functions practice works
1. Take the quiz: answer the exponential and logarithmic function questions at the top of the page.
2. Open the lesson (optional): review graphs, rules, and equation-solving strategies for exponentials and logs.
3. Retry: return to the quiz and apply exponential/logarithmic properties immediately.
What you will learn in the exponential & logarithmic functions lesson
Exponential function fundamentals & graphs
Definition: \(f(x)=ab^x\) where \(b>0\) and \(b≠ 1\)
Domain and range for \(b^x\) and key features like the horizontal asymptote
Increasing vs. decreasing behavior (growth vs. decay) and common transformations
Solving exponential equations
Rewrite to a common base and set exponents equal (when possible)
Use natural log \(\ln\) or log to solve equations like \(a^{kx}=c\)
Practice core forms such as \(2^{x+2}=16\), \(3^{2x-1}=9\), and \(e^x=1\)
Evaluate common logs and natural logs fast, like \(\log_{10}(1000)\) and \(\ln(e^2)\)
Translate between exponential form and log form confidently
Log rules, change of base & log equations
Log rules: product, quotient, and power rules for simplifying expressions
Change of base: \(\log_b(a)=\dfrac{\ln a}{\ln b}\) for calculators and simplification
Solve equations like \(\log_3(x-1)=2\) and \(\log_2(x)=-1\) and check domains
Back to the quiz
When you are ready, return to the quiz at the top of the page and keep practicing exponential and logarithmic functions.
⭐⭐⭐⭐⭐
📈
Exponential & Logarithmic
Step-by-step guide
Tap to open ->
Loading...
Exponential & Logarithmic Functions Lesson
1 / 8
Lesson Overview
Lesson overview
Purpose: Build a clear understanding of exponential and logarithmic functions so you can read and graph \(b^x\), use domain and range correctly, apply exponent rules, treat logarithms as inverse functions, simplify with log rules (product, quotient, power), use the change of base formula, and confidently solve exponential equations and solve logarithmic equations (with correct domain checks).
Success criteria
Recognize an exponential function (the variable is in the exponent), such as \(f(x)=3^x\) or \(f(x)=2^{x+1}\).
State the domain and range for basic exponential functions like \(b^x\) and \(e^x\).
Identify key graph features: the point \((0,1)\) for \(b^x\) and the horizontal asymptote \(y=0\).
Use exponent rules to rewrite expressions and solve equations with the same base.
Convert between log form and exponential form: \(\log_b(x)=y \iff b^y=x\).
Evaluate common values like \(\log_{10}(1000)\), \(\ln(1)\), and \(\ln(e^2)\).
Simplify using log rules and recognize when they apply (arguments must be positive).
Use change of base: \(\log_b(a)=\dfrac{\ln a}{\ln b}\).
Solve exponential equations by rewriting or taking logs (especially when bases don’t match).
Solve logarithmic equations and check solutions against domain restrictions.
Key vocabulary
Exponential function: a function where the variable is in the exponent, e.g. \(f(x)=b^x\) with \(b>0\), \(b≠ 1\).
Base: the number being raised to a power (for example, \(2\) in \(2^x\)).
Logarithm: \(\log_b(x)\) is the exponent you put on \(b\) to get \(x\).
Natural logarithm: \(\ln(x)=\log_e(x)\), the log base \(e\).
Common logarithm: \(\log(x)=\log_{10}(x)\), the log base \(10\).
Inverse functions: exponentials and logs undo each other: \(b^{\log_b(x)}=x\) (for \(x>0\)).
Asymptote: a line the graph approaches, like \(y=0\) for \(b^x\).
Quick pre-check
Pre-check 1: Which equation is equivalent to \(\log_{2}(8)=3\)?
Hint: \(\log_b(x)=y\) means \(b^y=x\).
Pre-check 2: What is \(\ln(1)\)?
Hint: \(\ln(1)=0\) because \(e^0=1\).
Exponential Functions
Exponential functions: graphs, domain, range, and asymptotes
Learning goal: Recognize exponential functions and state the key graph facts: domain, range, and the horizontal asymptote.
Key idea
A basic exponential function has the form \[ f(x)=b^x \quad \text{where } b>0 \text{ and } b≠ 1. \] Core facts you should know:
Domain: all real numbers \((-\infty,\infty)\).
Range: \((0,\infty)\) because \(b^x\) is always positive.
Key point: \(f(0)=b^0=1\), so the graph passes through \((0,1)\).
Asymptote: the graph approaches \(y=0\) but never reaches it.
Growth vs. decay: if \(b>1\) the function increases; if \(0<b<1\) it decreases.
Worked example
Example: What is the range of \(f(x)=3^x\)?
For every real \(x\), \(3^x>0\), so outputs are always positive. As \(x\to -\infty\), \(3^x\to 0\) (but never equals \(0\)). As \(x\to \infty\), \(3^x\to \infty\). So the range is: \[ (0,\infty). \]
Try it
Try it 1: What is the range of \(f(x)=e^x\)?
Hint: \(e^x\) is always positive and never equals \(0\).
Try it 2: Which statement is true about \(g(x)=\left(\tfrac{1}{2}\right)^x\)?
Hint: When \(0<b<1\), \(b^x\) represents exponential decay (decreasing).
Summary
For \(b^x\): domain \((-\infty,\infty)\), range \((0,\infty)\), asymptote \(y=0\).
Growth if \(b>1\); decay if \(0<b<1\).
Solving Exponential Equations
Solving exponential equations by rewriting bases and comparing exponents
Learning goal: Solve common exponential equations like \(2^{x+2}=16\), \(3^{2x-1}=9\), and \(e^x=1\) using reliable steps.
Key idea
If you can rewrite both sides using the same base \(b\) (with \(b>0\), \(b≠ 1\)), then: \[ b^{A}=b^{B}\ \Rightarrow\ A=B. \] This is the fastest method when the right-hand side is a clean power of the base.
Worked example
Example: Solve \(3^{2x-1}=9\).
Rewrite \(9\) as a power of \(3\): \[ 9=3^2. \] So \[ 3^{2x-1}=3^2 \Rightarrow 2x-1=2 \Rightarrow 2x=3 \Rightarrow x=\frac{3}{2}. \]
Try it
Try it 1: Solve \(2^{x+2}=16\).
Hint: \(16=2^4\). So set \(x+2=4\).
Try it 2: Solve \(e^x = 1\).
Hint: The exponent that gives \(1\) is \(0\): \(e^0=1\).
Summary
Rewrite both sides with the same base when possible.
If \(b^{A}=b^{B}\), then \(A=B\) (for \(b>0\), \(b≠ 1\)).
Logarithms Basics
Logarithms: definition, inverse relationship, and quick evaluation
Learning goal: Convert between log form and exponential form and evaluate common logarithms and natural logs accurately.
Key idea
A logarithm answers this question: “What exponent gives this result?” \[ \log_b(x)=y \iff b^y=x \] with the important condition \(x>0\). Two common special bases:
Common log: \(\log(x)=\log_{10}(x)\).
Natural log: \(\ln(x)=\log_e(x)\).
Worked example
Example: What is \(\log_{5}(125)\)?
Ask: “What power of \(5\) equals \(125\)?” Because \(5^3=125\), \[ \log_{5}(125)=3. \]
Try it
Try it 1: What is \(\log_{10}(1000)\)?
Hint: \(\log_{10}(1000)=3\) because \(10^3=1000\).
Try it 2: What is \(\ln(e^2)\)?
Hint: \(\ln\) and \(e^x\) undo each other: \(\ln(e^k)=k\).
Summary
\(\log_b(x)=y\) means \(b^y=x\) with \(x>0\).
\(\log_{10}\) is the common log; \(\ln\) is the natural log (base \(e\)).
Log Rules & Change of Base
Log rules for simplifying, plus the change of base formula
Learning goal: Simplify logarithms correctly using the product, quotient, and power rules, and use change of base to rewrite logs in terms of \(\ln\) or \(\log\).
Key idea
The three core log rules (for \(M>0\) and \(N>0\)) are:
If your calculator only has \(\ln\) and \(\log\), use change of base: \[ \log_b(a)=\frac{\ln a}{\ln b}=\frac{\log a}{\log b}. \]
Worked example
Example: Simplify \(\log_{3}(\sqrt{3})\).
Write \(\sqrt{3}=3^{1/2}\). Then apply the inverse idea: \[ \log_{3}(3^{1/2})=\frac{1}{2}. \]
Try it
Try it 1: Simplify using change of base: \(\log_2 7\).
Hint: \(\log_b(a)=\dfrac{\ln a}{\ln b}\).
Try it 2: Simplify \(\ln(2e)\).
Hint: \(\ln(AB)=\ln A+\ln B\) and \(\ln(e)=1\).
Summary
Use product/quotient/power rules only when arguments are positive.
Change of base: \(\log_b(a)=\dfrac{\ln a}{\ln b}\).
Solving Log Equations
Solve logarithmic equations and check domains
Learning goal: Solve log equations like \(\log_2(x)=4\) or \(\log_3(x-1)=2\) and avoid mistakes by checking the domain.
Key idea
The simplest log equation has the form \(\log_b(\text{expression})=c\). Convert to exponential form: \[ \log_b(A)=c \iff A=b^c. \] Domain rule: every log argument must be positive (for example, \(x-1>0\)).
Worked example
Example: Solve for \(x\): \(\log_3(x-1)=2\).
Convert to exponential form: \[ x-1=3^2=9 \Rightarrow x=10. \] Check the domain: \(x-1>0 \Rightarrow x>1\). The solution \(x=10\) is valid.
Try it
Try it 1: Solve \(\log_{2}(x)=4\).
Hint: \(\log_2(x)=4\) means \(2^4=x\).
Try it 2: Solve \(\log_{2}(x)= -1\).
Hint: \(\log_2(x)=-1\) means \(2^{-1}=x\).
Summary
Convert \(\log_b(A)=c\) to \(A=b^c\).
Always check the domain: every log argument must be \(>0\).
Natural Log & Solving
Use \(\ln\) to solve exponential equations when bases don’t match
Learning goal: Solve exponential equations using logarithms and understand why \(\ln\) is the standard tool for “bringing down” exponents.
Key idea
Logarithms are useful because they turn exponents into multiplication: \[ \ln\!\left(b^{g(x)}\right)=g(x)\ln(b). \] So when you can’t rewrite both sides with the same base, you can take \(\ln\) (or \(\log\)) of both sides and solve.
Worked example
Example: Solve \(e^{2x}=16\).
Take natural log of both sides: \[ \ln(e^{2x})=\ln(16). \] Use \(\ln(e^{2x})=2x\): \[ 2x=\ln(16)\Rightarrow x=\frac{\ln(16)}{2}. \] Since \(\ln(16)=\ln(4^2)=2\ln(4)\), you can simplify: \[ x=\ln(4). \]
Try it
Try it 1: Solve \(4^{x+2}=16\).
Hint: \(16=4^2\). So set \(x+2=2\).
Try it 2: Solve \(e^{2x}=16\).
Hint: Take \(\ln\) of both sides. You should get \(2x=\ln(16)\).
Summary
Taking \(\ln\) is powerful because \(\ln(b^{g(x)})=g(x)\ln(b)\).
Use logs to solve exponentials when you can’t match bases easily.
Applications & Big Picture
Why exponential and logarithmic functions matter
Learning goal: Connect exponentials and logs to real applications (growth, decay, and “solve for time” problems) — then finish with a final check.
Example: A quantity follows \(P(t)=500\cdot 2^{t}\). When does it reach \(4000\)?
Solve: \[ 500\cdot 2^t=4000 \Rightarrow 2^t=8. \] Because \(8=2^3\), we get \(t=3\).
Try it
Try it 1: Solve \(2^{x+1}=8\).
Hint: \(8=2^3\). So set \(x+1=3\).
Try it 2: Solve \(\log_{10} x=3\).
Hint: Convert to exponential form: \(10^3=x\).
Final recap
Exponential functions: \(f(x)=b^x\) with \(b>0\), \(b≠ 1\); domain \((-\infty,\infty)\), range \((0,\infty)\).
Logs are inverses: \(\log_b(x)=y \iff b^y=x\) (with \(x>0\)).
Common log: \(\log(x)=\log_{10}(x)\). Natural log: \(\ln(x)=\log_e(x)\).
Log rules: product, quotient, and power rules (arguments must be positive).
Change of base: \(\log_b(a)=\dfrac{\ln a}{\ln b}\).
To solve exponentials: match bases when possible; otherwise take logs to bring down exponents.
To solve logs: convert to exponential form and check the domain.
Next step: Close this lesson and try your quiz again. If you miss a question, reopen the book and review the page that matches the exponential or logarithmic skill you need.
Streak 5+
Streak 10+
Streak 15+
Streak 20+
Streak 25+