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Problem 16 - Entrance Test
If log_x (y) = 2 and log_y (z) = 3, then log_x (z) is equal to:
Correct: B
We are given two logarithmic equations:
1) log_x (y) = 2
2) log_y (z) = 3
We want to find log_x (z).
First, convert the logarithmic equations into exponential form:
From equation (1): log_x (y) = 2 means y = x^2.
From equation (2): log_y (z) = 3 means z = y^3.
Now, substitute the expression for y from the first exponential equation into the second exponential equation:
z = (x^2)^3.
Using the exponent rule (a^m)^n = a^(m*n):
z = x^(2*3)
z = x^6.
Finally, convert this back to logarithmic form with base x:
log_x (z) = log_x (x^6).
Using the logarithm property log_b (b^k) = k:
log_x (z) = 6.
Alternatively, using the change of base formula:
log_x (z) = log_x (y) * log_y (z) is not a general rule. The rule is log_a (c) = log_a (b) * log_b (c).
So, log_x (z) = log_x (y) * log_y (z) is correct given the base chaining.
log_x (z) = 2 * 3 = 6.