AP Calculus AB — Logarithms

This is the Calculus AB unit quiz for logarithms.

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Question 1
Which of the following is a simplification for:

$log⁡(3x)+log⁡(x)$

A
$log⁡(4x)$
B
$log⁡(3x^2 )$
C
$log⁡(2x)$
D
$log⁡(3)$
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:

$log⁡(3x)+log⁡(x)=log⁡(3x⋅x)=log⁡(3x^2)$
Question 2
Which of the following is a simplification for:

$log⁡(7x)+log⁡(3xy)-log⁡(3x)$

A
$log⁡(x^3 y)$
B
$log⁡(7x^2 y-3x)$
C
$log⁡(\frac{63x^2}{y^2})$
D
$log⁡(7xy)$
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:

$log⁡(\frac{21x^2 y} {3x}) = log⁡(7xy)$.
Question 3
Which of the following is equivalent to:

$log_2 (2^{log_2(2^7)} )$

A
$2$
B
$7$
C
$7^2$
D
$2^7$
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:

$log_2 (3) \cdot log_3 (4) \cdot log_4 (5) \cdot$ $\ldots \cdot log_{98} (99) \cdot log_{99} (100)$

A
$log_{99} (100)$
B
$log_{100} (2)$
C
$⁡\dfrac{log(2)}{2}$
D
$⁡\dfrac{2}{log(2)}$
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.

$\frac{log(3)}{log(2)} \cdot \frac{log(4)}{log(3)} \cdot \frac{log(5)}{log(4)} \cdot$ $... \cdot \frac{log(99)}{log(98)} \cdot \frac{log(100)}{log(99)}$ $= \frac{log(100)}{log(2)} = \frac{2}{log(2)}$
Question 5
Which of the following is equal to:

$log(z^2) + log(y^2) + log(x^2)$, where $log(x) + log(y) + log(z) = 8$

A
$8$
B
$16$
C
$64$
D
$128$
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 $2(8) = 16$.
Question 6
Which of the following is a simplification of:

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

A
$2 ln⁡(x) + 4$
B
$2 ln⁡(x) + e^4$
C
$ln(xe^4)$
D
$ln(8x)$
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:

$ln(x) + ln(x) – ln(x^2)$

A
$ln(x^4)$
B
$ln(x^2 + 2)$
C
$0$
D
$1$
Question 7 Explanation: 
The correct answer is (C). Using the sum and difference rules of logarithms, the expression can be rewritten as

$ln⁡(\frac{x^2}{x^2}) = ln(1) = 0$.
Question 8
Which of the following values of $x$ satisfies the equation:

$log_x(7) = 3$

A
$3^7$
B
$7^3$
C
$3^\frac{1}{7}$
D
$7^\frac{1}{3}$
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:

$log_{(x + 4)}(4) = 2$

A
$–2$
B
$0$
C
$2$
D
$4$
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 $x = –2$.
Question 10
Which of the following values of $x$ satisfies the equation:

$log_8(x^2 + 16x + 128) = 2$

A
$–8$
B
$–4$
C
$–2$
D
$0$
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 $x = –8$.
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