Exponential equation - logarithmisation

Question

is the transformation of this equation: $$9^x + 6^x = 2× 4^x$$ into this: $$\log_2 (9^x) + \log_2 (6^x)=\log_2 (2×4^x)$$ correct? I want to know because I really want to solve this equation.

Answer 1

HINT: write your equation in the form $$\left(\frac{3}{2}\right)^{2x}+\left(\frac{3}{2}\right)^x=2$$

Answer 2

No. One property of logarithms is that $\log(a) + \log(b) =\log(ab).$

So $\log(a + b)$ may not be reduced as you have done.

Answer 3

It's $f(x)=0$, where $$f(x)=\left(\frac{3}{2}\right)^{2x}+\left(\frac{3}{2}\right)^{x}-2.$$ We see that $f$ increases, which says that our equation has one root maximum.

But, $0$ is a root and we are done!

Your reasoning is wrong because $\log(a+b)$ is not always equal to $\log{a}+\log{b}.$