Hi, 







  , u = 5 

Answer by Theo(13342)   (Show Source): You can put this solution on YOUR website!
 you mean (log(125) to the base of u) + 8 = 14?

that would look like this:

log(u,125) = 

from that, i gather that this problem is:

2*log(u,125) + 8 = 14 which would look like:



if that's what it is, then the solution would be as follows:

subtract 8 from both sides of the equation to get:

2*log(u,125) = 14 - 8 which becomes:

2*log(u,125) = 6

divide both sides of this equation by 2 to get:

log(u,125) = 3

by the laws of logarithms log(b,x) = y if and only if b^y = x

based on this, your problem then becomes:

log(u,125) = 3 if and only if u^3 = 125

take the cube root of both sides of this equation to get:

u = (125)^(1/3)

the cube root of 125 should be 5 because 5^3 = 125.

if you didn't know that, you would use your calculator to solve for u to get:

u = (125)^(1/3) = 5

to confirm this is correct, use that value of u to solve the original equation.

the original equation is:

2*log(u,125) + 8 = 14

since u = 5, this becomes:

2*log(5,125) + 8 = 14

you can use the log base conversion formula to make this equivalent to:

2*log(10,125)/log(10,5) + 8 = 14

since the LOG function of your calculator already solves for logs to the base of 10, your equation becomes:

2*LOG(125)/LOG(5) + 8 = 14

you can now use your calculator to solve this equation to get:

14 = 14 confirming that the value of 5 for u is good.

the log conversion formula is:

log(b,c) = log(d,c)/log(d,b)

what this says is that the log of c to the base of b is equal to:

the log of c to the base of d divided by the log of b to the base of d.

c is the original number that you want to get the log of.

b is the original base.

d is the new base.

you take the log of the same number to the base of d (the new base) and divide that by the log of b (the old base) to the base of d (the new base).

in solving a problem like this, you need to know the basic definition of logarithms that states:

log(b,x) = y if and only if b^y = x

in symbolic notation, this would look like this:

 if and only if 

here's a reference that explains that in possibly a more elegant fashion than i could.

http://www.themathpage.com/aprecalc/logarithms.htm

 

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