Solve each equation

Question

log5 (4x-1) +log5 x=1

log3 4x- log3 7 = 2

Answer 1

1).

The logarithmic equation is log₅(4x - 1) + log₅(x) = 1.

Apply product property logarithm : .

log₅x(4x - 1) = 1

Apply formula : .

x(4x - 1) = 5

4x^2 - x - 5 = 0

By factoring by grouping.

4x^2 + 4x - 5x - 5 = 0

4x(x + 1) - 5(x + 1) = 0

(x + 1)(4x - 5) = 0

Apply zero product property.

x + 1 = 0 and 4x - 5 = 0

x = - 1 and x = 5/4.

We cannot have a log of a negative number, so x= - 1 must be rejected.

So, the solution is x = 5/4.

Answer 2

2).

The logarithmic equation is log₃(4x) - log₃(7) = 2.

Apply quotient property of logarithm : log_a(m) - log_a(n) = log_a(m/n).

log₃(4x/7) = 2

Apply formula : .

4x/7 = 3^2

4x/7 = 9

4x = 63

x = 63/4.

The solution is x = 63/4.