Answer :
The solutions of function log₆(x²+8) = 1 + log₆(x) are 2 and 4 which is correct option(D).
What are the properties of logarithms?
There are four basic properties of logarithms:
logₐ(xy) = logₐx + logₐy.
logₐ(x/y) = logₐx - logₐy.
logₐ(xⁿ) = n logₐx.
logₐx = logₓa / logₓb.
According to the problem, we will use some of the basic logarithmic properties,
log₆(x²+8) = 1 + log₆(x)
log₆(x²+8) - log₆(x) = 1
log₆(x²+8)/(x) = 1
Take antilog both side,
(x²+8)/(x) = 6
x²+8 = 6x
x²- 6x +8 = 0
Solving the quadratic equation,
x²- 4x -2x +8 = 0
x(x - 4) - 2(x - 4) = 0
(x - 4)(x - 2) = 0
Substitute the roots with zero,
x - 4 = 0 and x - 2 =0
x = 4 and x = 2
Hence, the solutions of function log₆(x²+8) = 1 + log₆(x) are 2 and 4.
Learn more logarithmic properties here:
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