Question

If loga 2=logb 8,show that a^3=b.

Answer 1

\[\large \log_a (2) = \log_b (8)\] Let's equate both of these logs to a number, \(x\), then:\[\large \log_a(2) = \log_b(8)=x\]\[\large \log_a(2)=x \implies a^x = 2\]cube both sides. \[\large (a^x)^3 = (2)^3 \implies a^{3x} = 2^3=8\]Same with \(\large \log_b(8)=x\)\[\large b^x = 8\]

Answer 2

One sec, checking over my work, Let me know if you are confused over anything up to this step though!

Answer 3

thanks again:)

Answer 4

Got it.

Answer 5

So you see how I cubed \(\large a^x\)to get it equal to 8?

Answer 6

I can take it's reverse to eliminate the number \(x\) I have as well!

Answer 7

Since \(\large a^{3x} = 8\) and \(b^x = 8\) i can equate them to eachother :) \[\large a^{3x} = b^x\]I want to prove that \(\large a^3 = b\) So i can reverse cancel the powers by taking the \(x\)th root :D\[\large \sqrt[x]{a^{3x}} = \sqrt[x]{b^x} \implies (a^{3x})^{1/x}=(b^x)^{1/x} \]\[\large \therefore ~a^3 = b\] :D