how do you work out 41.48 = 5 log(d) , where d is unknown. I seriously need help i'm about to give up here.

Question

it's for an astronomy lab that I really feel like a dumbass in.

Answer 1

divide both sides by 5 to get:
8.296 = log d

raise both sides of the equation to the power of ten.

10 ^ 8.296 = 10 ^ log d
10 ^ log d is just d because log and power of 10 cancel so:
d = 10^ 8.296 = 197696964

it is common to refer to the base 10 log with L O & G and not write the 10.
While logs of other bases are written differently to distiguish them from the ordinary log base 10.
Log to base e is commonly written as ln, and log any other base is written as log immediatly followed by a number (log#) with the # equaling a base other than 10.

Answer 2

Both the above answers are correct assuming one is working with the base-10 Log function.

If you are working with the Napier's Log function (more commonly known as Ln(x) := Log_e(x)) then the answer would be different. So a more accurate answer would be:

Let b be your base (e.g: 10 where Log_b(x) = Log_10(x) and here Log_10(100) = 2 etc), then:

41.48 = 5 * Log_b(d)
8.296 = Log_b(d)
b^8.296 = b^(Log_b(d))

now just like Log_10(10^2) = 2 here we have Log_b(b^x) = x, and so

d = b^8.296.

Now if b=10, d = 1976969664.01.....
or if b = e, d = 4007.8......

So just check the base ;-)

Answer 3

41.48 = 5 log(d) (meaning 41.48 is the result of 5 times the number that represents the number of times in which 10 is to be multiplied by itself to equal d)

By dividing both sides of the equation by 5, we get:

8.296 = log (d) (meaning 8.296 is equal to the number that represents the number of times in which 10 is to be multiplied by itself to equal d).

By raising both sides of the equation to be exponents of ten, we have:

10^8.296 = 10^log(d) (meaning 10^8.296 equals ten multiplied by itself in a number of times in which 10 is to be self-multiplied to equal d, hence d).

We can rewrite the above equation as follows:

10^8.296 = d.

By using a calculator, we can work out the result, which is 197696964.01118607721530209079508...

My solution is O.K., assuming log(d) means a logarithm to the base-10 of d.

Answer 4

41.48 = 5log(d)
log(d) = (41.48/5)
d = 10^(41.48/5)
d = about 1976969664.0112

just use inverse operations.