This is the Calculus AB unit quiz for 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: $log(7x)+log(3xy)-log(3x)$**

$log(x^3 y)$ | |

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

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

$log(7xy)$ |

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:

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

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

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:

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

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 $2log(z) + 2log(y) + 2log(x)$.

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 $2log(z) + 2log(y) + 2log(x)$.

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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