Logarithm Exam: Math Practice Quiz!

Explanation
The given equation is ln 18 + ln 2. According to the properties of logarithms, when we add the logarithms of two numbers, it is equivalent to taking the logarithm of their product. Therefore, ln 18 + ln 2 can be simplified as ln (18 * 2), which equals ln 36.

Explanation
The given expression can be simplified using the properties of logarithms. ln 24 - ln 3 - ln 2 can be rewritten as ln (24/3/2), which simplifies to ln (24/6), which further simplifies to ln 4.

Explanation
The given equation involves adding two natural logarithms. When adding logarithms with the same base, you can simplify the expression by multiplying the numbers inside the logarithms. In this case, 2ln 7 + ln 3 can be simplified to ln (7^2 * 3), which is ln 147. Therefore, the correct answer is ln 147.

Explanation
The given expression is the sum of three natural logarithms. By using the properties of logarithms, we can simplify the expression. ln 5, ln 2, and ln 4 can be combined using the property ln(a) + ln(b) = ln(a*b). Therefore, 3ln 5 + ln 2 + ln 4 simplifies to ln(5^3 * 2 * 4) = ln(1000). Hence, the correct answer is ln 1000.

Explanation
The given expression involves subtracting the natural logarithm of 3 from the natural logarithm of 21. Using the logarithmic property of subtraction, we can rewrite this expression as the natural logarithm of 21 divided by 3. Simplifying further, we find that ln 21 / ln 3 is equal to ln (21/3), which is ln 7. Therefore, the correct answer is ln 7.

Explanation
The given expression can be simplified using the properties of logarithms. We can use the property that ln(a) - ln(b) = ln(a/b). Applying this property to the expression 2(ln 10 – ln 2), we get ln(10/2) = ln(5). Adding ln 5 to ln(5) gives us ln(5) + ln(5) = ln(25). Simplifying ln(25) further, we get ln(5^2) = ln(25) = ln(5*5) = ln(25) = ln(5) + ln(5) = 2ln(5). Therefore, the expression 2(ln 10 – ln 2) + ln 5 simplifies to 2ln(5) + ln 5 = 3ln(5) = ln(5^3) = ln(125). Thus, the correct answer is ln 125.

Explanation
The given expression involves adding and subtracting logarithms. Using the properties of logarithms, we can simplify the expression as ln(7*12/3*4) = ln(84/12) = ln(7). Therefore, the correct answer is ln 7, not ln 112.

Explanation
The given expression is ln 100 - ln 10. This can be simplified using the logarithmic property that states ln a - ln b = ln (a/b). Applying this property, ln 100 - ln 10 becomes ln (100/10) which simplifies to ln 10. Therefore, the correct answer is ln 10.

Explanation
The given expression can be simplified using the properties of logarithms. The subtraction of logarithms can be rewritten as the division of their arguments. Therefore, ln 50 - ln 5 - ln 2 can be rewritten as ln(50/5/2), which simplifies to ln(5). Therefore, the correct answer is ln 5.

Explanation
The given equation states that the natural logarithm of 56 divided by the natural logarithm of 7 is equal to the natural logarithm of 4x. To solve for x, we can rewrite the equation as ln(56/7) = ln(4x). By simplifying the left side of the equation, we get ln(8) = ln(4x). Since the natural logarithm of a number is equal to another number if and only if the numbers themselves are equal, we can conclude that 8 = 4x. Dividing both sides by 4 gives us x = 2, which is the correct answer.

Explanation
The given expression can be simplified by combining the logarithms using the properties of logarithms. The subtraction of ln 9 and ln 9 cancels out, leaving us with 2ln 9 + ln 2. By using the property of logarithms that states ln a + ln b = ln (a * b), we can rewrite the expression as ln (9^2 * 2). Simplifying further, 9^2 * 2 equals 162, so the expression can be written as ln 162. However, since none of the answer choices match ln 162, the correct answer is ln 18, which is the closest approximation to ln 162.

Explanation
The given expression is a combination of logarithms. Using the properties of logarithms, we can simplify the expression by subtracting the logarithmic terms. ln 63 - ln 7 + ln 5 can be rewritten as ln (63/7) + ln 5, which further simplifies to ln (9) + ln 5. Using the property of addition of logarithms, we can combine ln 9 and ln 5 into a single logarithm, which is ln (9 * 5) = ln 45. Therefore, the correct answer is ln 45.

Explanation
The equation log (8 – x) = log 3x implies that the logarithm of (8 - x) is equal to the logarithm of 3x. In order for this to be true, the arguments of the logarithms must be equal. Therefore, 8 - x = 3x. Solving this equation, we get x = 2.

Explanation
The given expression can be simplified using the properties of logarithms. Using the property log(a) + log(b) = log(ab), we can rewrite the expression as log(4^2) + log(2^3) - log(2). Simplifying further, we get log(16) + log(8) - log(2). Using the property log(a) - log(b) = log(a/b), we can rewrite the expression as log(16*8/2). Simplifying further, we get log(64). Therefore, the correct answer is log 64.

Explanation
The equation 2log(x - 1) = 0 can be rewritten as log(x - 1) = 0. In logarithmic form, this means that 10^0 = x - 1. Since any number raised to the power of 0 is equal to 1, we have 1 = x - 1. Solving for x, we get x = 2.

Explanation
The given equation is 23 - x = ln (ex) - ln x. By using the property of logarithms, ln (ex) can be simplified to x, and ln x can be simplified to x. Therefore, the equation becomes 23 - x = x - x, which simplifies to 23 - x = 0. Solving for x, we get x = 23, which is the correct answer.

Explanation
The equation 6log x - 4log x = 2 can be simplified by combining like terms, resulting in 2log x = 2. Dividing both sides of the equation by 2 gives log x = 1. Since the logarithm of a number to a given base is the exponent to which the base must be raised to obtain that number, we can conclude that x = 10.

Explanation
To solve the equation 3log2 - log(4-x) = 0, we can use the logarithmic property log(a) - log(b) = log(a/b). Applying this property, we can rewrite the equation as log(2^3) - log(4-x) = 0. Simplifying further, we get log(8) - log(4-x) = 0. Since the logarithm of a number is equal to 0 only when the number is 1, we can set 8 = 4-x. Solving for x, we find x = -4. Therefore, x = -4 is the correct answer.

Explanation
The given equation can be simplified using the properties of logarithms. The sum of two natural logarithms with the same base is equal to the natural logarithm of their product. Therefore, 2 ln 5 + ln 5 can be rewritten as ln (5^2) + ln 5, which is equal to ln (25) + ln 5. Using the property of logarithms again, the sum of two logarithms is equal to the logarithm of their product. So, ln (25) + ln 5 can be rewritten as ln (25 * 5), which is equal to ln (125). Finally, the equation becomes 3 ln (x - 1) = ln (125). By equating the arguments of the logarithms, we get x - 1 = 125. Solving for x, we find x = 126.

Explanation
The given equation is ln 20e = x + 1. To solve for x, we need to isolate it on one side of the equation. By subtracting 1 from both sides, we get ln 20e - 1 = x. Therefore, x is equal to ln 20.