Question

If $\log_2 (x) + \log_4 (x) = 5$, what is $x$? 
 

• A. 2^10/2
• B. 2^10/3
• C. 2^11/3
• D. 2^10/7

Hint:

For equations mixing different log bases, rewrite all logs in terms of the same base and reduce to a simple algebraic equation.

Answer 1

The Correct Option is B

Step 1: Convert log₄(x) to base 2

Recall the base change formula:
log₄(x) = log₂(x) / log₂(4)
Since log₂(4) = 2, we get:
log₄(x) = log₂(x) / 2

Step 2: Rewrite the Equation

Substituting into the original equation:
log₂(x) + (log₂(x) / 2) = 5

Step 3: Combine Like Terms

Let y = log₂(x):
y + y/2 = 5
(3y / 2) = 5
3y = 10
y = 10/3

Step 4: Convert Back to x

Since y = log₂(x):
log₂(x) = 10/3
x = 2^10/3

Final Answer:

x = 2^10/3