Investment question using logarithm 8,000 dollars is invested in an account that yields 6% interest per year. After how many years will the account be worth $14 000, to the nearest half year, if the interest is compounded monthly?

Progress
$A = P (1 + i)^n$, so $14 000 = 8 000 (1 + 0.06)^n$, $14 000 = 8 000 (1.06)^n$, $1.75 = (1.06)^n$, $\log 1.75 / \log 1.06 = n$, and so $n = 9.6$ years. Am I correct?
OK.  If you have an annual percentage rate of $6\%$ compounded monthly, that means each month you add $\frac{6\%}{12}$ of the investment to itself.  

So each month the investment is multiplied by by $1.005$.  ($\text{Investment plus Investment}\times0.005$).

So after $x$ months, your investment is $8000\times(1.005)^x$  Therefore, you need to solve the following equation:$$8000\times(1.005)^x=14000$$

to get the number of months it will take to get to \$14000.  I see in your comment above that you know how to solve this. You will have to divide by 12 to get years...
Halfway between 9 and 10 years. Set a formula multiplying the base by 1.06 per year, on the compounding amount.