heather_johnson
heather_johnson 12h ago โ€ข 0 views

Test Your Knowledge: Solving Exponential Equations with Logarithms and Different Bases

Hey there, math whiz! ๐Ÿ‘‹ Exponential equations with different bases throwing you for a loop? Don't sweat it! This study guide + quiz will help you nail those problems using logarithms. Let's get started! ๐Ÿค“
๐Ÿงฎ Mathematics

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katie.sutton Dec 27, 2025

๐Ÿ“š Quick Study Guide

  • ๐Ÿ”ข Exponential Equation Form: An exponential equation takes the form $a^x = b$, where $a$ is the base, $x$ is the exponent, and $b$ is the result.
  • ๐Ÿ“ Logarithm Definition: The logarithm (base $a$) of $b$ is the exponent to which $a$ must be raised to equal $b$. Mathematically, if $a^x = b$, then $\log_a(b) = x$.
  • ๐Ÿงฎ Change of Base Formula: $\log_a(b) = \frac{\log_c(b)}{\log_c(a)}$, where $c$ can be any convenient base, typically 10 or $e$ (natural logarithm). This is crucial when dealing with different bases.
  • ๐Ÿ’ก Solving Strategy: To solve $a^x = b$, take the logarithm of both sides (using the same base) or apply the change of base formula. Then, isolate $x$.
  • โž• Logarithm Properties: Useful properties include: $\log(mn) = \log(m) + \log(n)$, $\log(\frac{m}{n}) = \log(m) - \log(n)$, and $\log(m^p) = p \cdot \log(m)$.
  • ๐Ÿงช Natural Logarithm: Denoted as $\ln(x)$, this is the logarithm with base $e$ (Euler's number, approximately 2.718). It is often used in calculus and other advanced math.

Practice Quiz

  1. What is the value of $x$ in the equation $3^x = 27$?
    1. A) 2
    2. B) 3
    3. C) 4
    4. D) 9
  2. Solve for $x$: $5^x = 125$
    1. A) 25
    2. B) 3
    3. C) 5
    4. D) 1
  3. Solve for $x$: $2^x = 10$. (Round to two decimal places)
    1. A) 3.32
    2. B) 1.00
    3. C) 0.30
    4. D) 4.00
  4. What is the value of $x$ in the equation $4^x = 16^{2}$?
    1. A) 2
    2. B) 4
    3. C) 6
    4. D) 8
  5. Solve for $x$: $7^x = 49$
    1. A) 14
    2. B) 2
    3. C) 7
    4. D) 3
  6. Solve for $x$: $6^x = 216$
    1. A) 2
    2. B) 36
    3. C) 3
    4. D) 4
  7. What is the value of $x$ in the equation $9^x = 81$?
    1. A) 3
    2. B) 6
    3. C) 9
    4. D) 2
Click to see Answers
  1. B) 3
  2. B) 3
  3. A) 3.32
  4. B) 4
  5. B) 2
  6. C) 3
  7. D) 2

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