Qfyilyi

$$left(frac 43right)^x=frac{8sqrt3}{9}$$

I've been trying to solve this question. I know that the answer is $3/2$ by guess and check, but I want to know how to do it by actually solving it.

I think that perhaps the best way of tackling this problem is not through logarithms, but rather by raising both sides of your equation to another power:
$$
left(frac{4}{3}right)^x = frac{8sqrt{3}}{9} quad Rightarrow quad left(frac{4}{3}right)^{2x} = left(frac{8sqrt{3}}{9}right)^2
$$
Now things are looking up! The right hand side (RHS) is now equal to $64/27$, which we can much more readily identify as $4^3/3^3$. But if I need to cube $4/3$ then what value is $x$? Well, $2x=3$ implies $x=3/2$!

Generally speaking, logarithms with non-integer bases are hard to work with; when possible, instead just continue to work with exponents.

$$frac{8sqrt3}{9}=frac{2^3cdot 3^{0.5}}{3^2}=frac{2^3}{3^{1.5}}=left(frac{2^2}{3}right)^{1.5}$$

Note that the function $f(x)=left( frac43right)^x$ is an increasing function.

Hence $x=1.5$.

Alternatively, taking logarithm on both sides.

Since you have tagged the question as a logarithmic question, let us try to approach it that way. Using the basic properties of log:
$$
(frac{4}{3})^x = frac{8 sqrt{3}}{9} \
implies x log{4} - x log{3} = log{8 sqrt{3}} - log{9} \
implies x log{4} - x log{3} = log{(2 times 4)} + log{3^{0.5}} - log{3^2} \
implies log{4^x} - log{3^x} = log{4^{0.5}} + log{4^1} + log{3^{0.5}} - log{3^2} \
implies log{4^x} - log{3^x} = log{4^{1.5}} - log{3^{-0.5}} - log{3^2} \
implies log{4^x} - log{3^x} = log{4^{1.5}} - log{3^{1.5}} \
implies log{(frac{4}{3})^x} = log{(frac{4}{3})^{1.5}}
implies x log{frac{4}{3}} = 1.5 log{frac{4}{3}} \
implies x = 1.5 = frac{3}{2} space square
$$

I'm not sure how deep you want to go into this with the rules of logarithms, but the simple answer is that

$$
log_b (x^n) = nlog_b (x).
$$

Thus, if you take the logarithm (say base 10) of both sides, then you have,

$$
log_{10} (4/3)^x = log_{10} (8sqrt 3/9)
$$
This leads to,
$$
x log_{10} (4/3) = log_{10} (8sqrt 3/9).
$$

If you divide both sides by $log_{10}(4/3)$ you isolate x and find your answer of 3/2.