Question

Log4^2(x)-3log4(x)=3^(log3(4))

Answer 1

(log3)(log3+logx)=(log4)(log4+logy)and(logx)(log4)=(logy)(log3)

logy=log4log3logx

(log3)(log3+logx)=(log4)(log4+log4log3logx)

(log3−(log4)2log3)logx=(log4)2−(log3)2

logxlog3=−1

logx=−log3=log(1/3), so that x=1/3.

Answer 2

To solve the equation log₄²(x) - 3log₄(x) = 3^(log₃(4)), we can simplify the terms and apply some logarithmic properties.

Let's break it down step by step:

1. Rewrite 3^(log₃(4)) using the change of base formula: 3^(log₃(4)) = 4^(log₄(3))

2. Apply the power rule of logarithms to rewrite the equation: log₄²(x) - log₄(x)³ = 4^(log₄(3))

3. Combine the logarithmic terms on the left side using the quotient rule of logarithms: log₄²(x) - 3log₄(x) = 4^(log₄(3))

4. Apply the power rule of logarithms to rewrite the left side: log₄(x²) - log₄(x³) = 4^(log₄(3))

5. Combine the logarithmic terms on the left side using the quotient rule of logarithms: log₄(x²/x³) = 4^(log₄(3))

6. Simplify the left side by dividing x² by x³: log₄(1/x) = 4^(log₄(3))

7. Apply the logarithmic property to rewrite log₄(1/x): log₄(1) - log₄(x) = 4^(log₄(3))

8. Simplify log₄(1) to 0: 0 - log₄(x) = 4^(log₄(3))

9. Rearrange the equation: -log₄(x) = 4^(log₄(3))

10. Apply the exponential property to both sides: 4^(-log₄(x)) = 3

11. Rewrite the left side using the reciprocal property: 1/(4^log₄(x)) = 3

12. Apply the logarithmic property to rewrite 4^log₄(x): 1/x = 3

13. Multiply both sides by x to isolate x: x = 1/3

Therefore, the solution to the equation log₄²(x) - 3log₄(x) = 3^(log₃(4)) is x = 1/3.