Can someone see if I got these correct?
1. Solve log5 x = 3.
x = 125
2. log3 (x - 4) > 2.
x > 13
3. Evaluate log7 49.
log7 49 = 2
4. Solve log3 x = 4.
x = 81
5. The graph of a logarithmic function y = 3^x is a reflection of x = 3^y over the line y = 0.
False
6. Evaluate log7 7^5.
=5
7. Does log5 (12p) = log5 12 + log5 p?
Yes
8. solve the equation, log4 z + log4 8 = log4 24. z = 16
9. Solve In (x - 6)< - 3.
x < 6 + ^e(-3)
10. Write an equivalent expression for x if e^x+3 = 6.
x = -3 + In 6
#8. 8z = 24, so z=3 (see question 7!)
#10 correct, if you meant e^(x+3) = 6
as you wrote it, x = ln 3
The others are all good.
2logx + log2 =2
To verify if the given answers are correct, let's go through each question and explain how to solve it:
1. Solve log5 x = 3.
To solve this equation, you need to raise the base (5) to the power of the exponent (3) to find the value of x. In this case, 5^3 = 125, so x = 125. The answer provided is correct.
2. log3 (x - 4) > 2.
To solve this logarithmic inequality, you need to isolate the logarithm first. Rewrite the inequality as an exponential equation: 3^(log3(x - 4)) > 3^2. Simplifying this, you get x - 4 > 9. Solving for x, x > 13. The answer provided is correct.
3. Evaluate log7 49.
To evaluate this logarithm, you need to determine the power to which the base (7) should be raised to equal the argument (49). In this case, 7^2 = 49, so log7 49 = 2. The answer provided is correct.
4. Solve log3 x = 4.
To solve this equation, you need to raise the base (3) to the power of the exponent (4) to find the value of x. In this case, 3^4 = 81, so x = 81. The answer provided is correct.
5. The graph of a logarithmic function y = 3^x is a reflection of x = 3^y over the line y = 0.
This statement is false. The graph of y = 3^x is an exponential function, not a logarithmic function. The reflection statement is incorrect.
6. Evaluate log7 7^5.
When a logarithm has the same base as its argument, the result is simply the exponent. In this case, log7 7^5 = 5. The answer provided is correct.
7. Does log5 (12p) = log5 12 + log5 p?
Yes, this equation holds true. According to the logarithmic property of the product, loga (mn) = loga m + loga n. Applying this property to the equation, log5 (12p) = log5 12 + log5 p. The answer provided is correct.
8. Solve the equation, log4 z + log4 8 = log4 24.
To solve this equation, you can use the logarithmic property of the sum. According to loga (mn) = loga m + loga n, you can rewrite the equation as one logarithm: log4 (z * 8) = log4 24. Simplifying further, z * 8 = 24. Solving for z, z = 24/8 = 3. The answer provided is incorrect; the correct answer is z = 3.
9. Solve ln (x - 6) < -3.
To solve this inequality, you need to isolate the natural logarithm expression. ln (x - 6) < -3 can be rewritten as e^(ln(x - 6)) < e^(-3). Simplifying, x - 6 < e^(-3). Adding 6 to both sides, x < 6 + e^(-3). The answer provided is incorrect; the correct answer is x < 6 + e^(-3).
10. Write an equivalent expression for x if e^(x+3) = 6.
To find an equivalent expression for x, you need to isolate the exponential expression. Taking the natural logarithm of both sides gives ln(e^(x+3)) = ln(6), which simplifies to x + 3 = ln(6). Subtracting 3 from both sides, x = ln(6) - 3. The answer provided is correct.
Please let me know if you have any further questions or if anything needs clarification.