If $10^a = 8$ and $10^b=12$ and $10^x = 6$ what is in a and b

Question

The title pretty much sums it up. If we had an equation $10^a = 8$ and $10^b=12$ what is $x$ in terms of $a$ and $b$ if $10^x =6$. I feel like the answer is simple, but yet I only got so far.

Thank you

Answer 1

$$10^b=12$$

$$ \implies 10^{3b}=12^3 \tag1$$

And ,

$$10^a=8 \tag2$$

Dividing equation $(1)$ by $(2)$ :

$$\frac{10^{3b}}{10^a}=\frac{12^3}{8}=\left(\frac{12}{2}\right)^3=6^3 =10^{3x}$$

$$\implies 10^{3b-a}=10^{3x}$$

Since the base is same, exponents must also be same, that is :

$$3b-a=3x$$

Answer 2

Taking logarithm base $10,$ $$a=\log8=3\log2$$

$$b=\log12=2\log2+\log3\implies\log3=b-\dfrac{2a}3$$

$$x=\log2+\log3=\dfrac a3+b-\dfrac{2a}3$$