Solve the logarithmic equation. $\log ( \sqrt [ 3 ] { x } ) = \sqrt { \log ( x ) }$
A) x = 1
B) x = 1,000,000,000, x = 1
C) x = 10,000,000, x = $-$ 1
D) x = 100,000,000

Question

Quizplus · Accepted Answer

First, we can simplify the left side of the equation using the property $\log_b \left( a^n \right) = n \log_b (a)$. We get:

$$\log \left( \sqrt [ 3 ] { x } \right) = \frac{1}{3} \log (x)$$

Using substitution $y= \log(x)$, we can rewrite the original equation as:

$$\frac{1}{3}y = \sqrt{y}$$

Squaring both sides gives:

$$\frac{1}{9}y^2 = y$$

This simplifies to:

$$y(y-9) = 0$$

Therefore, either $y=0$ or $y=9$.

Substituting these values back in for $y$, we get $x=1$ and $x=10^9$. So the answer is B.

Solve the Logarithmic Equation log⁡(x3)=log⁡(x)\log ( \sqrt [ 3 ] { x } ) = \sqrt { \log ( x ) }log(3x​)=log(x)​

Solve the Logarithmic Equation $\log ( \sqrt [ 3 ] { x } ) = \sqrt { \log ( x ) }$