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JEE Main · 2021 · 26 Feb 2021 · Shift 1 · Basic Mathematics

The number of solution of the

lo g_{4} (x - 1) = lo g_{2} (x - 3)

is ____

Answer: 1

Solution

Convert to the same base. Using

lo g_{4} (x - 1) = \frac{l o g _{2} ( x - 1 )}{2}

:

\frac{l o g _{2} ( x - 1 )}{2} = lo g_{2} (x - 3)

, so

lo g_{2} (x - 1) = 2 lo g_{2} (x - 3) = lo g_{2} (x - 3)^{2}

.

This gives

x - 1 = (x - 3)^{2}

, so

x - 1 = x^{2} - 6 x + 9

, hence

x^{2} - 7 x + 10 = 0

.

Factoring:

(x - 5) (x - 2) = 0

, so

x = 5

or

x = 2

.

Check domain:

x > 3

(for

lo g_{2} (x - 3)

to be defined). Only

x = 5

is valid.

Therefore, there is 1 solution.

JEE Main · 2021 · 26 Feb 2021 · Shift 1 · Integer answer

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