I need the value of d.

This was given as a formula to find a distance. m-M=5log(d/10)

I went ahead and plugged in the given information, which led to it being 1.08=log(d/10) and I am still trying to find d, the distance. Any better?

Yes, much better!  there is no ^ here. So using a property of logarithms (see the Formulary) we can write:

1.08 = log(d) - log(10)

Now log to the base 10 of 10 is just 1 (ie log₁₀(10)=1) so 

1.08 = log(d) - 1 

Add 1 to both sides

2.08 = log(d)

Now raise both sides to the power 10

10^2.08 = d because 10^log(d) = d

So:

$${\mathtt{d}} = {{\mathtt{10}}}^{{\mathtt{2.08}}} \Rightarrow {\mathtt{d}} = {\mathtt{120.226\: \!443\: \!461\: \!741\: \!290\: \!6}}$$

 or d is approximately 120.

No thanks to anyone else, I think I've got the answer.      1.08=log^d/10

This is a log base 10. That means I can cancel out the 10's which leaves me with 1.08=logd.

Now I just need to rewrite this as an exponential equation. 10^1.08=d.

d is approximately 12.

I'm afraid neither your question nor your answer makes any mathematical sense.  What do you mean by log^d? Do you just mean log(d)?  Do you mean log(d/10)? Do you mean log(d)/10? Do you mean something else?  

And just because your log might be to base 10, that doesn't mean you can just cancel it with another 10.  

What I mean is 1.08= log with (d/10) as its exponent. I need to find the value of d. I pretty sure you could cancel the base 10 with the bottom 10 in its exponent. I hope the question is more comprehensible now. Please, can anyone help?

This still isn't meaningful!

What does the expression log^(d/10) mean? You have to have the logarithm of something before you can start raising to a power.  

Perhaps you mean log(d^1/10) in which case your equation would be 1.08 = log(d^1/10), which is equivalent to 1.08 = (1/10)log(d), so log(d) = 10.8 so d = 10^10.8. 

or do you mean 1.08 = log(d/10) in which case 1.08 = log(d) - log(10) so 1.08 = log(d) - 1 so log(d) = 2.08 so d = 10^2.08. 

or do you mean 1.08 = (log(d))^1/10, in which case 1.08¹⁰ = log(d) so d = 10^{(1.08^10)}?

You could have log(m^d/10) or log(m)^d/10, but you would then need to specify the value of m in order to find d.

This was given as a formula to find a distance. m-M=5log(d/10)

I went ahead and plugged in the given information, which led to it being 1.08=log(d/10) and I am still trying to find d, the distance. Any better?

thankyou very much, this was very helpful!