maths logs

Question

1. Write each of the following more simply (i.e. not involving logs)

log_^a a²
0.1^logx

b).solve the following equations for x, giving your answers to 4 significant figures:

x^-1.73 = 4.82

2^-x = 0.1

2. In a right angled triangle ABC, angle B = 90^o and the sides a and c have lengths

a=1.7
c=2.8

how long is side b?

Answer 1

For 1 a) Recall that exponents contained within log can be pulled out front. For example, in base 10, Log (1000) = Log(10³) = 3 Log(10) = 3. Apply this idea to get your solution.

 

For 1 b)  You need to get that exponent out of there, So apply ln to both sides and you get,

x^-1.73 = 4.82 → -1.73 ln(x)= ln(4.82) → ln (x)= ln(4.82)/-1.73

To undo the natural log we must place our values in the exponent of e. So we get ,

e ^{ln (x)}= x = e ^{ln(4.82)/-1.73}

 

For 2) Since we have a right triangle we should be able to make use of the Pythagorean theorem. Normally it is a²+b²=c² when the right angle is located at C. Since the right angle is located at B you must swap b and c. Which will give you,

a²+ c²=b². You should be able to solve it from here.