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# Logarithms and Simultaneous Equations

1. Given that x and y are both positive, solve the simultaneous equations

log(xy) =7

log(x/y) =1

Answers given: x = 10000
y = 1000

2. log (p - q +1) =0

log (pq) + 1 = 0

Show that p = q = 1/square root 10

3. Solve the following equation for x:

2^2x + 1 - 9 x 2^x + 4 = 0

Answer given: -1, 2

4. Example of exam question

Solve the pair of simultaneous equations

log(x + y) = 0

2 log x = log (y - 1)

#### Solution Preview

I assume you are familiar with the log laws and definition of log (see appendix)

Questions 1 and 2 are in the attachments, it was easier to just scan them.

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Q3. Solve the following equation for x:

2^(2x + 1) - 9 x 2^x + 4 = 0 (this was sloppy, you must use brackets to be clear.)

Answer given: -1, 2

SOLUTION: This is a quadratic in 2^x.

2 (2^x)^2 - 9 (2^x) +4 = 0

Factorise to get
( 2(2^x) - 1 )( 2^x - 4 ) = 0

2(2^x) - 1 = 0 or 2^x - 4 = 0

2(2^x) = 1 or 2^x = 4

2^x = 1/2 ...

#### Solution Summary

Logarithms and simultaneous equations are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

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