0

# Calculus logarithm question

Log x + log (x+15) = 2

How is this solved?

### 2 Answers by Expert Tutors

Phillip R. | Top Notch Math and Science Tutoring from Brown Univ GradTop Notch Math and Science Tutoring from...
1
I use two rules which you should memorize.

First, if A and B are numbers greater than zero, then log A + log B = log AB.
This rule combines two logs into one.

Second, if log X = Y, then by definition, 10Y = X
This rule changes a log equation into an exponential equation.

So starting with log X + log (X + 15) = 2, we use the first rule to get

log [X(X + 15)] = 2

using the second rule, we get

102 = X(X + 15)

Now we can solve for X

X2 + 15X - 100 = 0

(X - 5) (X + 20) = 0

Therefore X = 5 or X = -20

The answer X = -20 is eliminated because if we use it in the original equation we get log (-20) which is undefined because the domain of the log function is X > 0.

So we have one answer X = 5
Joseph B. | Texas Certified Math and Science TeacherTexas Certified Math and Science Teacher
5.0 5.0 (1155 lesson ratings) (1155)
0
The composite of a function and its inverse is just the original argument (argument means input) and so your problem is solved by understanding that the inverse of a log function is the exponential function with the same base. The base of the common log is 10 (log X means log10x) therefore it's inverse is 10x.

In summary:

10log(x) = x

We therefore solve the problem as follows:

log x + log (x+15) = 2

log [x(x+15)] = 2 (the sum of two common logs is the common log of the product of their arguments)

10{log [x(x+15)]} = 102  (since left and right side of equation are equal we plug each into the 10x function)

x(x+15) = 100

x^2 + 15x = 100

x^2 + 15x - 100 = 0 (factor the quadratic)

(x+20) (x-5) = 0

By zero product rule:
x+20 = 0 or x-5 = 0

so

x = -20 or x = 5

since the log function doesn't exist for negative numbers (log -20 doesn't exist) the answer is x = 5.