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Log graphs and exponential

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Two variables X and Y, are connected by the law Y=a( power of X ). The graph of log4Y against x is a straight line passing through the origin and the point A(6,3). Find the value of a.

Y axis is log ( base 4 )Y

X axis = X
(edited 7 years ago)
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mik1a
10
If the log4(y)-x graph is a straight line then you can write: log4(y) = mx + c (think of log4(y) as a new variable for the time being).

Now you want to find m and c, and you can do this using the knowledge that this line passes (0,0) and (6,3). In other words you know m and c have to be numbers such that 0 = m*0 + c and 6 = m*3 + c, which are the two conditions for the line to pass through these points. So you now solve these simultaneous equations to find m and c:

c = 0 from the first equation;
m = 2 from the second equation.

So you know now that log4(y) = 2x.

Now solve. You can get rid of the log by taking four to the power of each side of the equation (i.e. 4^LHS = 4^RHS):
4^(log4(y)) = 4^(2x)
y = 4^(2x)

And we know from indices rules that 4^(2x) = 4^2^x = (4^2)^x = 16^x

So a = 16.


summary:
- if two things plotted give a straight line, they must be related by a y=mx+c type equation.
- if you know the line y=mx+c passes through the point (a,b) then you know that b must equal m*a+c
- logs are hard, get rid of them to solve things, and remember your indices rules.
(edited 7 years ago)

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