This is our AP Calculus AB unit test on logarithms.

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Question 1 |

**Which of the following is a simplification for: $\log(3x)+\log(x)$**

$\log(4x)$ | |

$\log(3x^2 )$ | |

$\log(2x)$ | |

$\log(3)$ |

Question 1 Explanation:

The correct answer is (B). The addition rule for logarithms is:

$\log(a)+\log(b)=\log(ab)$

In this case, we see that:

$\log(3x)+\log(x)=$ $\log(3x⋅x)=\log(3x^2)$

$\log(a)+\log(b)=\log(ab)$

In this case, we see that:

$\log(3x)+\log(x)=$ $\log(3x⋅x)=\log(3x^2)$

Question 2 |

**Which of the following is a simplification for: $\log(7x)+\log(3xy)-\log(3x)$**

$\log(x^3 y)$ | |

$\log(7x^2 y-3x)$ | |

$\log\left(\frac{63x^2}{y^2}\right)$ | |

$\log(7xy)$ |

Question 2 Explanation:

The correct answer is (D). The first two logarithms can be combined using the addition rule. So:

$\log(7x)+\log(3xy)=$ $\log(21x^2 y)$

Since the third logarithm is being subtracted from the other two, we must divide $21x^2 y$ by the argument of $\log(3x)$. This results in:

$\log\left(\frac{21x^2 y} {3x}\right) = \log(7xy)$

$\log(7x)+\log(3xy)=$ $\log(21x^2 y)$

Since the third logarithm is being subtracted from the other two, we must divide $21x^2 y$ by the argument of $\log(3x)$. This results in:

$\log\left(\frac{21x^2 y} {3x}\right) = \log(7xy)$

Question 3 |

**Which of the following is equivalent to:**$\log_2 (2^{\log_2(2^7)} )$

$2$ | |

$7$ | |

$7^2$ | |

$2^7$ |

Question 3 Explanation:

The correct answer is (B). To solve this problem quickly, start with the innermost logarithm. Since $\log_2 2^7 = 7$, the upper expression can be rewritten as $\log_2 (2^7 )$. This logarithm can be further simplified as $\log_2 (2^7 ) = 7$.

Question 4 |

**Which of the following is equal to:**$\log_2 (3) \cdot \log_3 (4) \cdot \log_4 (5) \cdot$ $\ldots \cdot \log_{98} (99) \cdot \log_{99} (100)$

$\log_{99} (100)$ | |

$\log_{100} (2)$ | |

$\dfrac{\log(2)}{2}$ | |

$\dfrac{2}{\log(2)}$ |

Question 4 Explanation:

The correct answer is (D). The log change of base formula is: $\log_a (b) = \frac{\log(b)}{\log(a)}$

We can apply this formula to a few elements in this series and look for a pattern.

$\frac{\log(3)}{\log(2)} \cdot \frac{\log(4)}{\log(3)} \cdot \frac{\log(5)}{\log(4)} \cdot$ $... \cdot \frac{\log(99)}{\log(98)} \cdot \frac{\log(100)}{\log(99)}$ $= \frac{\log(100)}{\log(2)} = \frac{2}{\log(2)}$

We can apply this formula to a few elements in this series and look for a pattern.

$\frac{\log(3)}{\log(2)} \cdot \frac{\log(4)}{\log(3)} \cdot \frac{\log(5)}{\log(4)} \cdot$ $... \cdot \frac{\log(99)}{\log(98)} \cdot \frac{\log(100)}{\log(99)}$ $= \frac{\log(100)}{\log(2)} = \frac{2}{\log(2)}$

Question 5 |

**Which of the following is equal to:**$\log(z^2) + \log(y^2) + \log(x^2)$,

**where**$\log(x) + \log(y) + \log(z) = 8$

$8$ | |

$16$ | |

$64$ | |

$128$ |

Question 5 Explanation:

The correct answer is (B). The logarithmic exponent rule states that: $\log(x^n) = n × \log(x)$

We can rewrite the sum of the three logarithms in the given expression as: $2\log(z) + 2\log(y) + 2\log(x)$

Factoring out $2$ gives: $2(\log(x) + \log(y) + \log(z))$

Since the problem stated that $\log(x) + \log(y) + \log(z) = 8$, substituting this value into the expression gives: $2(8) = 16$

We can rewrite the sum of the three logarithms in the given expression as: $2\log(z) + 2\log(y) + 2\log(x)$

Factoring out $2$ gives: $2(\log(x) + \log(y) + \log(z))$

Since the problem stated that $\log(x) + \log(y) + \log(z) = 8$, substituting this value into the expression gives: $2(8) = 16$

Question 6 |

**Which of the following is a simplification of:**$\ln(x^2) + \ln(e^4)$

$2 \ln(x) + 4$ | |

$2\ln(x) + e^4$ | |

$\ln(xe^4)$ | |

$\ln(8x)$ |

Question 6 Explanation:

The correct answer is (A). The first logarithm can be simplified to $2 \ln(x)$ using the logarithmic exponent rule. The second logarithm can be simplified to $4$, since the natural logarithm is base $e$. The sum of these two logarithms is $2 \ln(x) + 4$.

Question 7 |

**Which of the following is equivalent to:**$\ln(x) + \ln(x) − \ln(x^2)$

$\ln(x^4)$ | |

$\ln(x^2 + 2)$ | |

$0$ | |

$1$ |

Question 7 Explanation:

The correct answer is (C). Using the sum and difference rules of logarithms, the expression can be rewritten as:

$\ln(\frac{x^2}{x^2}) = \ln(1) = 0$

$\ln(\frac{x^2}{x^2}) = \ln(1) = 0$

Question 8 |

**Which of the following values of $x$ satisfies the equation:**$\log_x(7) = 3$

$3^7$ | |

$7^3$ | |

$3^\frac{1}{7}$ | |

$7^\frac{1}{3}$ |

Question 8 Explanation:

The correct answer is (D). From the definition of a logarithm, we know that $x^3 = 7$. Therefore, the value of $x$ is the cube root of $7$, or $7^\frac{1}{3}$.

Question 9 |

**Which of the following values of $x$ satisfies the equation:**$\log_{(x + 4)}(4) = 2$

$−2$ | |

$0$ | |

$2$ | |

$4$ |

Question 9 Explanation:

The correct answer is (A). From the definition of logarithm: $(x + 4)^2 = 4$

Expanding and simplifying gives: $x^2 + 8x − 12 = 0$

Using the quadratic formula, the possible solutions for $x$ are:

$\dfrac{–8±4}{2}$

This means $x = −6$ or $x = −2$. Since log bases cannot be negative, $x = −6$ cannot be a valid answer.

Thus, the only possible value for $x$ that satisfies the equation is: $x = −2$

Expanding and simplifying gives: $x^2 + 8x − 12 = 0$

Using the quadratic formula, the possible solutions for $x$ are:

$\dfrac{–8±4}{2}$

This means $x = −6$ or $x = −2$. Since log bases cannot be negative, $x = −6$ cannot be a valid answer.

Thus, the only possible value for $x$ that satisfies the equation is: $x = −2$

Question 10 |

**Which of the following values of $x$ satisfies the equation:**$\log_8(x^2 + 16x + 128) = 2$

$-8$ | |

$-4$ | |

$-2$ | |

$0$ |

Question 10 Explanation:

The correct answer is (A). From the definition of logarithm: $x^2 + 16x + 128 = 64$

Subtracting $64$ from both sides gives $x^2 + 16x + 64 = 0$. This quadratic can be factored as $(x + 8)^2 = 0$.

The only possible solution for $x$ that satisfies this equation is: $x = −8$

Subtracting $64$ from both sides gives $x^2 + 16x + 64 = 0$. This quadratic can be factored as $(x + 8)^2 = 0$.

The only possible solution for $x$ that satisfies this equation is: $x = −8$

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