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Exam Question stuck: Logs @ Sine/cos/tan rules

I am stuck on these questions and don't know what to do... I am not sure about the starting steps
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Reply 1
Avatar for mike256
mike256
2
Here are some hints:

For question 7, notice that for a product of two brackets to equal zero, either of the brackets must equal zero. And remember that sin θ / cos θ = tan θ.

For question 9a, you have to use the trapezium rule, 1/2 h [f(0) + f(1) + 2(f(0.25) + f(0.5) + f(0.75))], where f(x) = log(x^2 + 1).

9b is a simple transation by a vector. The rule is that if y in the original equation becomes y - k in the new equation, the curve has been translated by vector [0, k].

9c is mostly use of log rules, which are
log(ab) = log a + log b
log(a/b) = log a - log b
log x^n = n log x.

Let me know where in particular you are stuck.
Reply 2
Avatar for Freddy12345
Freddy12345
OP
Original post by mike256
Here are some hints:

For question 7, notice that for a product of two brackets to equal zero, either of the brackets must equal zero. And remember that sin θ / cos θ = tan θ.

For question 9a, you have to use the trapezium rule, 1/2 h [f(0) + f(1) + 2(f(0.25) + f(0.5) + f(0.75))], where f(x) = log(x^2 + 1).

9b is a simple transation by a vector. The rule is that if y in the original equation becomes y - k in the new equation, the curve has been translated by vector [0, k].

9c is mostly use of log rules, which are
log(ab) = log a + log b
log(a/b) = log a - log b
log x^n = n log x.

Let me know where in particular you are stuck.


I am stuck because the log10 is putting me off and I can only do simple log work...
Reply 3
Avatar for mike256
mike256
2
Original post by Freddy12345
I am stuck because the log10 is putting me off and I can only do simple log work...


log10 means logarithm to base 10, i.e.
log⁡10x=y\log_{10} x = y
means that
10y=x10^y = x

It behaves like all other logarithms.

Which part of the question are you stuck on?
Reply 4
Avatar for Freddy12345
Freddy12345
OP
Original post by mike256
log10 means logarithm to base 10, i.e.
log⁡10x=y\log_{10} x = y
means that
10y=x10^y = x

It behaves like all other logarithms.

Which part of the question are you stuck on?


First part says use trapezium rule, does that mean I have to integrate first, or use trapezium rule straightaway?

Also Part c (ii) 4 mark question, both equations are the same, so I don't understand what's going on there.
Reply 5
Avatar for mike256
mike256
2
Original post by Freddy12345
First part says use trapezium rule, does that mean I have to integrate first, or use trapezium rule straightaway?

Also Part c (ii) 4 mark question, both equations are the same, so I don't understand what's going on there.


The trapezium rule is for estimating the integral, so you don't integrate the equation first (we wouldn't be able to integrate logs with only Core 2 knowledge).

For part (c)(ii), the second equation has a 1 added to it. Because you need to show that the transformation is a stretch, you want to rearrange the equation

y = 1 + 2log10 x

so that it is in the form

y = 2log10(kx), where k is some stretch factor.

Start by rewriting it as

y = log10 10 + 2log10 x, because 1 = log1010.

The difficult part is to realise that you need to somehow combine the terms log1010 + 2log10x. You could use the rule that log a + log b = log(ab), but that only works if the coefficients of the logs are the same, so you need to rewrite log10 10 as 2 log (something). Hint: use the fact that 2log a = log a2.

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