Solve the equation for the indicated unknown.

Question

Solve the equation for the indicated unknown. Write your answers exactly and then, where appropriate, approximate to four decimal places. 

 

log(m-1)-log(m+1)=-1

Answer 1

In solving the problem, I make the assumption that here log means "logarithm base 10", not base e. 

 

We'll start off with by using the following logarithm rule:

log(a/b) = log(a) - log(b)

 

log(m-1) - log(m+1) = -1.  Apply the rule:

log[(m-1)/(m+1)] = -1.

 

To solve for m, we must first undo the logarithm on the left-hand side.  This is accomplished by exponentiation, using the same base as the logarithm.  In this case, the base is 10.  We treat both sides of the equation as the exponent of 10:

 

10^{log[(m-1)/(m+1)]} = 10^-1.  Logarithm and exponentiation undo eachother on the left.  On the right, 10^-1 = 1/10:

(m-1)/(m+1) = 1/10.  From here, it's all algebra. Cross-multiply:

(1)(m+1) = 10(m-1).  Continue to solve for m:

m + 1 = 10m - 10

9m = 11

m = 11/9