
Help with logs please
We just started doing some work on logs and i cant work out this problem, i think its quite easy but im crap at these.
Two curves:
y=a^x and y=2b^x
Prove that x coord of the point of intersection is 1 / (log(of base2)Alog(of base2)B)
Help would be appreciated

y=a^x and y=2b^x
y = a^x (1)
y = 2(b^x) (2)
At their point of intersection, the two logarithmic curves have the same coordinates.
So, y from (1) = y from (2)
a^x = 2(b^x) (3)
Since we are looking for the xcoordinate of the intersection, then we solve for x.
Take the logs, (log to the base 10), of both sides of (3),
x*log(a) = log(2) +x*log(b)
Collect the xterms,
x*log(a) x*log(b) = log(2)
x[log(a) log(b)] = log(2)
Divide both sides by [log(a) log(b)],
x = log(2) / [log(a) log(b)] (4)
(4) is on base 10.
Since you want x in base 2, then we transform the logs in 4 into logs to the base 2.
You still remember how to change bases for logarithms?
log(base a) to log(base b).
log(base a)(N) = [log(base b)(N)] / [log(base b)(a)] ***
So,
>>>log(base 10)(2)
= [log(base 2)(2)] / [log(base 2)(10)]
= 1 / [log(base 2)(10)]
>>>log(base 10)(a)
= [log(base 2)(a)] / [log(base 2)(10)]
>>>log(base 10)(b)
= [log(base 2)(b)] / [log(base 2)(10)]
Substitute those into (4),
x = log(2) / [log(a) log(b)] (4)
x = log(base 10)(2) / [log(base 10)(a) log(base 10)(b)]
x = {1 / log(base 2)(10)} / {[log(base 2)(a) / log(base 2)(10)] [log(base 2)(b) / log(base 2)(10)]}
Combine the denominator into one fraction only,
x = {1 / log(base 2)(10)} / {[log(base 2)(a) log(base 2)(b)] / log(base 2)(10)}
Reverse the denominator, to multiply it to the numerator,
x = {1 / log(base 2)(10)} * {log(base 2)(10) / [log(base 2)(a) log(base 2)(b)]}
The log(base 2)(10) cancels out,
x = 1 / [log(base 2)(a) log(base 2)(b)] answer.
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Or, from (3), we can take the logs to the base 2 directlywithout passing through the logs to the base 10.
a^x = 2(b^x) (3)
log(base 2)(a^x) = log(base 2)(2 b^x)
x*log(base 2)(a) = log(base 2)(2) +x*log(base 2)(b)
x*log(base 2)(a) = 1 +x*log(base 2)(b)
x*log(base 2)(a) x*log(base 2)(b) = 1
x[log(base 2)(a) log(base 2)(b)] = 1
x = 1 / [log(base 2)(a) log(base 2)(b)] answer.

Ah ok!
Thanks very much for your help