Thread: Logarithm problem with terminology I dont quite get

Ok, this is a word problem in the book I am working in, and I am kind of stuck, because I dont completely understand all the terms, so I was looking for some guidance. The word problem is as follows:

The intensity L(x) of light x meters under sea level is given as:

L(x)=L₀a^x

where L₀ = L(0) is the light intensity at sea level. A diver has found that the intensity is reduced to half at 6 meters under sea level. The diver cannot work without the aid of a lamp when the intensity is under 1/10 of the value of sea level. Calculate how deep the diver can go without the need for a lamp to complete his work.

Ok, so I know that L(x) is just a function. However, I have not yet encountered the terms where they turn it into a subscript. (L₀) So I thought to myself that since that has to equal 100% intensity, I can just change it out with 100, and then the problem becomes:

100*a^x

The value I am looking for is 1/10 the value of 100, so 10. So the equation would be:

100*a^x=10

However, there are two variables here. a and x. I understand what the x is, as it is how many meters down, and it is the variable that I am trying to solve. But what is with the a? I have a feeling that my lack of understanding of the terms here is locking me, and that the L₀ shouldnt actually be set to 100, although it seems logical to me.

Can anyone point me in the right direction? Thank you.

The a is a constant that tells us something about the clarity of the water.

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For example:

With a=1 you would have , impossibly clear water.

With a=1/2 you have or , for each metre you descend the light intensity is halved.

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In your question you are told that when x=6m the light intensity is half the surface intensity.

Sub it in your equation .

Can you solve that to find the value of a?