Logarithm Problem Set 1

1. Solve for x:

log_x64=2

The equation logx64=2 can be rewritten as x^2=64. Taking the square root of both sides, we find that x=±8. However, since the logarithm function is only defined for positive values of x, the only valid solution is x=8.

Explanation

The equation logx64=2 can be rewritten as x^2=64. Taking the square root of both sides, we find that x=±8. However, since the logarithm function is only defined for positive values of x, the only valid solution is x=8.

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2. Evaluate:
log₄16

The given expression is asking for the logarithm of 416 base 2. To evaluate this, we need to find the exponent to which 2 must be raised to get 416. Since 2^8 = 256 and 2^9 = 512, we know that 416 is between these two powers of 2. Therefore, the answer is 8.

Explanation

The given expression is asking for the logarithm of 416 base 2. To evaluate this, we need to find the exponent to which 2 must be raised to get 416. Since 2^8 = 256 and 2^9 = 512, we know that 416 is between these two powers of 2. Therefore, the answer is 8.

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3. Solve for x:

log₄x=3

To solve the equation log4x = 3, we need to rewrite it in exponential form. Since the base of the logarithm is 4, we can rewrite it as 4^3 = x. Simplifying this equation gives us x = 64. Therefore, the value of x that satisfies the equation is 64.

Explanation

To solve the equation log4x = 3, we need to rewrite it in exponential form. Since the base of the logarithm is 4, we can rewrite it as 4^3 = x. Simplifying this equation gives us x = 64. Therefore, the value of x that satisfies the equation is 64.

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4. Evaluate:
log₆(1/36)

The given expression is evaluating the logarithm of 1/36 to the base 6. The logarithm of a number is the exponent to which the base must be raised to obtain that number. In this case, we need to find the exponent that 6 must be raised to in order to get 1/36. Since 6 raised to the power of -2 equals 1/36, the answer is -2.

Explanation

The given expression is evaluating the logarithm of 1/36 to the base 6. The logarithm of a number is the exponent to which the base must be raised to obtain that number. In this case, we need to find the exponent that 6 must be raised to in order to get 1/36. Since 6 raised to the power of -2 equals 1/36, the answer is -2.

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5. Solve for x:

log₃x=-2
Enter answer as a fraction or as a decimal to the thousandths place.

The correct answer is 1/9, 0.111, .111.

To solve the equation log3x = -2, we need to rewrite it in exponential form. Since the base of the logarithm is 3, we can rewrite it as 3^-2 = x. Simplifying this, we get x = 1/9. Therefore, the answer is 1/9.

Additionally, 1/9 can be expressed as a decimal, which is 0.111 when rounded to the thousandths place. Hence, both 0.111 and .111 are also correct answers.

Explanation

The correct answer is 1/9, 0.111, .111.

To solve the equation log3x = -2, we need to rewrite it in exponential form. Since the base of the logarithm is 3, we can rewrite it as 3^-2 = x. Simplifying this, we get x = 1/9. Therefore, the answer is 1/9.

Additionally, 1/9 can be expressed as a decimal, which is 0.111 when rounded to the thousandths place. Hence, both 0.111 and .111 are also correct answers.

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6. Log_(1/2)8=x

The equation log base 1/2 of 8 equals x can be rewritten as 1/2 raised to the power of x equals 8. To find the value of x, we need to determine what exponent, when 1/2 is raised to that power, gives us 8. In this case, x equals -3 because 1/2 raised to the power of -3 is equal to 8.

Explanation

The equation log base 1/2 of 8 equals x can be rewritten as 1/2 raised to the power of x equals 8. To find the value of x, we need to determine what exponent, when 1/2 is raised to that power, gives us 8. In this case, x equals -3 because 1/2 raised to the power of -3 is equal to 8.

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7.

Evaluate:

Enter answer as an improper fraction or a decimal

The given question asks to evaluate the expression 3/2. The answer is both 3/2 and 1.5 because they represent the same value. 3/2 is an improper fraction, where the numerator (3) is greater than the denominator (2). On the other hand, 1.5 is a decimal representation of the same value. Both answers are correct and equivalent.

Explanation

The given question asks to evaluate the expression 3/2. The answer is both 3/2 and 1.5 because they represent the same value. 3/2 is an improper fraction, where the numerator (3) is greater than the denominator (2). On the other hand, 1.5 is a decimal representation of the same value. Both answers are correct and equivalent.

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8. What is 5^log₅x equivalent to?

The expression 5log5x can be simplified using the logarithmic property, which states that log_b(b^a) = a. In this case, the base of the logarithm is 5 and the exponent is x. Since the base and the logarithm are the same, the expression simplifies to x.

Explanation

The expression 5log5x can be simplified using the logarithmic property, which states that log_b(b^a) = a. In this case, the base of the logarithm is 5 and the exponent is x. Since the base and the logarithm are the same, the expression simplifies to x.

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9. Which of the following is the inverse of the function f(x) = log_(1/2)x?

A. f^-1(x) = 2^x
B. f^-1(x) = (1/2)^x
C. f^-1(x) = x²
D. f^-1(x) = x^(1/2)
E. That function has no inverse.

A

B

C

D

E

The correct answer is B because to find the inverse of a function, we switch the x and y variables and solve for y. In this case, we have f(x) = log(1/2)x. To find the inverse, we switch x and y to get x = log(1/2)y. Then, we solve for y by exponentiating both sides with base 2, which gives us 2^x = y. Therefore, the inverse function is f-1(x) = (1/2)x.

Explanation

The correct answer is B because to find the inverse of a function, we switch the x and y variables and solve for y. In this case, we have f(x) = log(1/2)x. To find the inverse, we switch x and y to get x = log(1/2)y. Then, we solve for y by exponentiating both sides with base 2, which gives us 2^x = y. Therefore, the inverse function is f-1(x) = (1/2)x.

10. Which of the following is the correct inverse of the function f(x) = 4^x ?

A. f^-1(x) = x⁴
B. f^-1(x) = x ^1/4
C. f^_-1(x) = log₄x
D. f^-1(x) = log_(1/4)x
E. This function has no inverse.

A

B

C

D

E

The correct inverse of the function f(x) = 4x is f-1(x) = log4x. The inverse function undoes the original function, so when we apply the inverse function to the output of the original function, we should get back the input. In this case, if we apply f-1(x) = log4x to the output of f(x) = 4x, we get log4(4x) = x. Therefore, C is the correct answer.

Explanation

The correct inverse of the function f(x) = 4x is f-1(x) = log4x. The inverse function undoes the original function, so when we apply the inverse function to the output of the original function, we should get back the input. In this case, if we apply f-1(x) = log4x to the output of f(x) = 4x, we get log4(4x) = x. Therefore, C is the correct answer.