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Chapter 8: Problem 5

Solve (a) \(\log 2 x=1.5\) (b) \(\log (3 x+1)=2.1500\) (c) \(\log \left(x^{2}+3\right)=2.3671\) (d) \(4 \log (5 x-6)=-0.8000\)

Short Answer

Expert verified
A: The solutions are: (a) x = 15.81 (b) x = 46.75 (c) x = ±15.1 (d) x = 1.326192

Step by step solution

01

(a) Solve \(\log 2x = 1.5\)

Step 1: Convert the Equation to Exponential Form Our equation is \(\log 2x = 1.5\). Convert it into exponential form: \(2x = 10^{1.5}\). Step 2: Solve for x \(2x = 10^{1.5}\) can be simplified to \(2x = 31.62\). To solve for x, divide by 2: \(x = \frac{31.62}{2} = 15.81\).
02

(b) Solve \(\log (3x + 1) = 2.15\)

Step 1: Convert the Equation to Exponential Form Our equation is \(\log (3x + 1) = 2.15\). Convert it into exponential form: \(3x + 1 = 10^{2.15}\). Step 2: Solve for x \(3x + 1 = 10^{2.15}\) can be simplified to \(3x + 1 = 141.25\). Subtract 1 from both sides to get \(3x = 140.25\). Now divide by 3: \(x = \frac{140.25}{3}= 46.75\).
03

(c) Solve \(\log (x^2 + 3) = 2.3671\)

Step 1: Convert the Equation to Exponential Form Our equation is \(\log (x^2 + 3) = 2.3671\). Convert it into exponential form: \(x^2 + 3 = 10^{2.3671}\). Step 2: Solve for x \(x^2 + 3 = 10^{2.3671}\) can be simplified to \(x^2 + 3 = 231.08\). Subtract 3 from both sides to get \(x^2 = 228.08\). Now take the square root of both sides: \(x = \pm\sqrt{228.08} = \pm 15.1\).
04

(d) Solve \(4\log(5x - 6) = -0.8\)

Step 1: Isolate the Logarithm Our equation is \(4\log(5x - 6) = -0.8\). Divide by 4 on both sides to isolate the logarithm: \(\log(5x - 6) = -0.2\). Step 2: Convert the Equation to Exponential Form Now, we can convert it into exponential form: \(5x - 6 = 10^{-0.2}\). Step 3: Solve for x \(5x - 6 = 10^{-0.2}\) can be simplified to \(5x - 6 = 0.63096\). Add 6 to both sides to get \(5x = 6.63096\). Now, divide by 5: \(x = \frac{6.63096}{5} = 1.326192\).

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