AP Pre Calculus Free Response Question Logarithm

Functions g and h

The functions g and h are given by

g(x) = log₃(x + 4)

h(x) = 2 cos x

(i) Solve g(x) = 2 for values of x in the domain of g.

Since g(x) = log₃(x + 4), we set:

log₃(x + 4) = 2

Rewrite the logarithmic equation in exponential form:

x + 4 = 3²

x + 4 = 9

x = 5

The domain of g is x + 4 > 0, so x > -4. Since 5 is in the domain, the solution is:

x = 5

(ii) Solve h(x) = 1 for values of x in the interval [0, 2π].

Since h(x) = 2 cos x, we set:

2 cos x = 1

Divide both sides by 2:

cos x = 1/2

On the interval [0, 2π], cosine equals 1/2 at:

x = π/3 and x = 5π/3

Therefore, the solutions are:

x = π/3, 5π/3

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