Hi,
I wonder if anyone can help me with the following please?

Two variables x and y are connected by the law y = a^x. The graph of log 4 y against x is a straight line passing through the origin and the point (6,3). Find the value of 'a'.

First I converted the y = a^x into log a y = x, but got nowhere substituting (6, 3) into the equation.
Next I tried to take logs of both sides:
3 = a^6 log 3 = 6 log a log a = log 3/6 but this gave me the answer 1.2 (the answer is 2)

Can anyone direct me to which method to use to solve this?

Thanks

jiten55

RATHER A TRICKY QUESTION!!

(You should have typed log 4 y as log y (base 4) to avoid confusion. I thought you meant log (4y) and had to try both alternatives)

Suppose coordinate system is (Y,x)

Then the equation of the line passing through the two given points is Y = x/2

Let Y = Log (base 4) y

Equation of line is Y = x/2

y = a^x

(Base for Log is 4 all through)

log y = x log a ie Y = x log a (Y = log y)

Y = x log a shoud be identical to Y = x/2

Hence log a = 1/2

Base is 4

a = 4^1/2 = 2

Let me know if any step not clear.

eawoodall

the answer is wrong.
y = a ^ x.
substitute.
0 = a ^ 0. wrong.
0 <> a ^ 0 = 1.
any real to zero power equals one!
they must have typoed someplace.
3 = a ^ 6?

coordinate system is (x,y).
maybe they again got order wrong. and they forgot about a ^0 = 1.

jiten55

(See my Solution given earlier)

The important point is that this is a case of coordinate transformation.

In many real-life problems, variables (eg x, y) do not have a desired functional relation while transformation of a variable (in this case Log y (base 4)) can result in a desirable functional relation (eg Straight line, Linear, etc.)

xxzzy

I am now very confused. Is Jiten 55's response correct or am I to understand from eawoodall that there is a typo in the question?
And what does the statement "You must spread some Reputation around before giving it to jiten55 again" as a rating mean?

jiten55

Main thing for you is to make sure you understand every step of a solution.

NOTE the fact that the graph is not of y against x BUT of Log y (base 4) against x. The question does NOT say that (0, 0) lies on y = a^x.

If you understand every step of an answer, and every step makes Mathematical sense to you, then you can trust the answer!

As I wrote, if you have difficulty in understanding any step, do ask for assistance and I will try to clarify.

In Mathematics, if every step is correct then the conclusion has got to be correct!

This is unlike subjects like history where it is often a matter of opinions.

As regards some "comment" on "ratings", again it is entirely up to you. If YOU like
a response, you have every right to give it a good rating - because someone took time to help you.

BUT NEVER REST until you understand each step of a Solution. Otherwise you will never be good at Mathematics!

jiten55

ANOTHER Solution (Perhaps easier)

ALL logs are taken with base 4

y = a^x (Equation 1)

Hence log y = x log a (Equation 2)

(0, 0) lies on the graph of Log y against x

Hence, when x =0, log y = 0 (This condition is automatically satisfied by equation 2)

(6, 3) lies on the graph of Log y against x.

Hence, when x = 6, Log y = 3

Substitute in Equation 2:

3 = 6 log a

log a = 1/2

(Base is 4)

a = 4^(1/2) = 2

Hence a = 2

xxzzy

Thanks Jiten 55 that explanation was totally clear.
Sorry I couldn't give you a high rating but that statement' You must spread etc .....' was there.
Thank you again. I hope I can come back to you with a question in the future !

eawoodall

the explanation of the question was wrong.

i answered what they asked.

i am glad you saw through what they asked to see what they wanted.

but i was not wrong. you are wrong for presuming so jiten55.

what i did was take your equation 1, and make it into this:

log (base a) y = x (equation 3)
which has to be mathematically true if equation 1 is true, always!
they said it went through origin, which it cannot because the origin is (0,0).
and for no base a is 1=0.

secondly they said it was a straight line. log are not straight lines.

the related equation you showed is a straight line.
they did not ask the right type of question, because they do not understand that they are dealing with more than one line. a log line, and a related line that is straight.

glad you were able to fix their problem with their understanding what the question should have been.

but of course we are not suppose to do people homework problems for them, just find a way to show them how to do it themselves. and hopefully your example will show them how to do similar ones. personally i think the problem needs to be reworded.

jiten55

XXZZY, You are very welcome.

Your questions have been very interesting.

I love questions that are NOT very easy to solve!