If log_a(4) = 2 and log_a(16) = x, what is the value of x?

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Question: If log_a(4) = 2 and log_a(16) = x, what is the value of x?

Options:

Correct Answer: 4

Solution:

log_a(16) = log_a(4^2) = 2 * log_a(4) = 2 * 2 = 4.

If log_a(4) = 2 and log_a(16) = x, what is the value of x?

If log_a(4) = 2 and log_a(16) = x, what is the value of x?

Step 1: We know that log_a(4) = 2. This means that a^2 = 4.
Step 2: We want to find log_a(16). Notice that 16 can be written as 4^2 (because 4 * 4 = 16).
Step 3: We can use the property of logarithms that says log_a(b^c) = c * log_a(b).
Step 4: Apply this property: log_a(16) = log_a(4^2) = 2 * log_a(4).
Step 5: Substitute the value of log_a(4) into the equation: log_a(16) = 2 * 2.
Step 6: Calculate the result: 2 * 2 = 4.
Step 7: Therefore, x = log_a(16) = 4.

Logarithmic Properties – Understanding the properties of logarithms, particularly the power rule which states that log_b(m^n) = n * log_b(m).
Change of Base – Recognizing how to manipulate logarithmic expressions to find unknown values using known logarithmic values.