Chapter 8: Problem 12

Solve (a) \(2 \ln (3 x-10)=8.5\) (b) \(\log \left(x^{3}+1\right)=2.4\) (c) \(3 \log 4 x-8=0\) (d) \(\frac{\ln 5 x}{2}=1.6\)

Short Answer

Expert verified

Question: Solve the following logarithmic equations for x: a) \(2 \ln (3 x-10)=8.5\) b) \(\log \left(x^{3}+1\right)=2.4\) c) \(3 \log 4 x-8=0\) d) \(\frac{\ln 5x}{2}=1.6\) Answer: a) \(x \approx 7.7\) b) \(x \approx 3.4\) c) \(x \approx 37.3\) d) \(x \approx 8.0\)

Step by step solution

1. Isolate the natural logarithm

Divide both sides of the equation by 2 to isolate the natural logarithm: \(\ln(3x-10)=4.25\)

2. Apply exponentiation

Apply exponentiation on both sides using e as the base (since it's a natural logarithm): \(3x-10=e^{4.25}\)

3. Solve for x

Move the constant to the other side and divide by 3 to solve for x: \(x=\frac{e^{4.25}+10}{3}\)

4. Evaluate the expression

Evaluate the expression for x using a calculator to get the approximate value: \(x\approx 7.7\)

5. Solve the second equation

Raise both sides of the equation to the power of 10 (since it's a common logarithm): \(x^3+1=10^{2.4}\)

6. Solve for x

Subtract 1 from both sides, then take the cube root to solve for x: \(x=\sqrt[3]{10^{2.4}-1}\)

7. Evaluate the expression

Evaluate the expression for x using a calculator to get the approximate value: \(x\approx 3.4\)

8. Solve the third equation

Add 8 to both sides of the equation: \(3\log 4x=8\)

9. Apply exponentiation

Divide both sides by 3 and raise both sides of the equation to the power of 10: \(4x=10^{\frac{8}{3}}\)

10. Solve for x

Divide both sides by 4 to solve for x: \(x=\frac{10^{\frac{8}{3}}}{4}\)

11. Evaluate the expression

Evaluate the expression for x using a calculator to get the approximate value: \(x\approx 37.3\)

12. Solve the fourth equation

Multiply both sides of the equation by 2: \(\ln 5x = 3.2\)

13. Apply exponentiation

Apply exponentiation on both sides using e as the base: \(5x = e^{3.2}\)

14. Solve for x

Divide both sides of the equation by 5 to solve for x: \(x=\frac{e^{3.2}}{5}\)

15. Evaluate the expression

Evaluate the expression for x using a calculator to get the approximate value: \(x\approx 8.0\) So the solutions for the given exercises are: a) \(x\approx 7.7\) b) \(x\approx 3.4\) c) \(x\approx 37.3\) d) \(x\approx 8.0\)

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Solve (a) \(2 \ln (3 x-10)=8.5\) (b) \(\log \left(x^{3}+1\right)=2.4\) (c) \(3 \log 4 x-8=0\) (d) \(\frac{\ln 5 x}{2}=1.6\)

Short Answer

Step by step solution

1. Isolate the natural logarithm

2. Apply exponentiation

3. Solve for x

4. Evaluate the expression

5. Solve the second equation

6. Solve for x

7. Evaluate the expression

8. Solve the third equation

9. Apply exponentiation

10. Solve for x

11. Evaluate the expression

12. Solve the fourth equation

13. Apply exponentiation

14. Solve for x

15. Evaluate the expression

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