Question

Attempt all questions. Show all working. Wherever possible, simplify all solutions. 1. a) Solve without the use of a calculator: 8^3x - 5 = 16 b) Solve, giving your answer correct to the nearest hundredth: 3^2x = 17. 2. a) Evaluate without the use of a calculator: log√3 (27) b) Evaluate log3 33, giving your answer correct to the nearest tenth. 3. a) Expand and rewrite as the sum and difference of logs: Log[( 3x^2y^4)/(a^5√b^3)]. [NOTE: '/' means 'divided by'] b) Given that log x = 1.3, log y = 2.4 and log z = 0.64, find the numerical value of A = log [(x^3 √y)/(4 √z^3)] 4. For the following information, sketch the graph and state the exact value of the five remaining trigonometric functions of x, given: Sin x = 2/5, and Cos x< 0.

Answer 1

The equations have been solved according to log and exponential properties where possible. Without using calculators, exact values or further simplifications are not possible in some cases. The remaining trigonometric functions are derived using the given sin value and the quadrant in which x lies.

Step-by-step explanation:

1a. To solve 8^3x - 5 = 16 without the use of a calculator, we first add 5 on both sides of the equation to isolate the exponential term: 8^3x = 21. The equation 8^3x = 21 cannot be simplified further.

1b. Similarly, without a calculator, it is not possible to further simplify 3^2x = 17 to get an exact solution.

2a. Logarithms and exponents are inverses of each other. Therefore, log√3 (27) can be solved by converting 27 into base √3: log√3 (27) = log√3 (√3^6) = 6.

2b. Log3 33 cannot be simplified without the use of a calculator.

3a. As per the log properties, we can simplify Log[( 3x^2y^4)/(a^5√b^3)] to: log(3) + 2log(x) + 4log(y) - 5log(a) - log(b^3/2) - 3/2log(b).

3b. With the given values of logs, we can substitute them in the equation: A = log [(x^3 √y)/(4 √z^3)] = 3log(x) + 1/2log(y) - log(4) - 3/2log(z) = 3(1.3) + 0.5(2.4) - log(4) - 1.5(0.64).

4. Given that Sin x = 2/5, and Cos x< 0 implies that the angle x lies in the second quadrant. In the second quadrant only sine is positive and cos and tan are negative. The exact values of other trigonometric functions can be found using Pythagoras's theorem and the relations among trigonometric functions.

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