Exponential and Logarithmic Equations

Main Topic – Exponential and Logarithmic Equations   

Sub topic - Solve logarithmic equations

Solve the equation log(x) + log(x + 4) = log (32) for x

A. x = -8

B. x = 4 

C. x = 4 and x = -8

D. x = 14

Answer: B

Rationale

Rules of Logarithm are used.

First step is to use the addition rule of logarithm.

log (a) + log (b) = log (a*b)

Therefore, the equation becomes

log [x (x + 4)] = log(32)

Second step is to get rid of logarithm from both sides.

So, x (x + 4) = 32

=> x2 + 4x = 32

Third step is, to solve this quadratic equation.

x2  + 4x – 32 =0

(x - 4) (x + 8) = 0

x = 4 and x = -8

Next step is, to identify the correct solution.

x = -8 cannot be the solution, since log(x) is not defined for negative values.

Therefore x = 4 is the only solution.

Answer option A is incorrect: log(x) is not defined for negative values.

Answer option C is incorrect: log(x) is not defined for negative values.

Answer option D is incorrect: Use the addition rule of logarithm and then solve the quadratic equation by using factor method or quadratic formula.