Question

log (x+21)+logx=2 solve the equation

Answer 1

x = 4 is the correct answer.

Answer 2

The sum of logs is the log of the product.

Law of logarithms:

$\log_b x + \log_b y = \log_b xy$

Apply the law above to the left side of the equation.

$\log (x + 21) + \log x = 2$

$\log [x(x + 21)] = 2$

$\log (x^2 + 21x) = 2$

Now use the definition of log.

$\log_b x = y \Leftrightarrow b^y = x$

$x^2 + 21x = 10^2$

$x^2 + 21x - 100 = 0$

$(x + 25)(x - 4) = 0$

$x + 25 = 0~~~\lor~~~x - 4 = 0$

$x = -21~~~\lor ~~~x = 4$

x = -21 must be discarded because log (x + 21) would become log (-21) which is not defined.

Solution: x = 4