Question

checkpoint 1.31
solve $ln(x^{3}) - 4ln(x)=1$.

Explanation:

Step1: Use logarithm property

Using the power - rule of logarithms $\ln(a^b)=b\ln(a)$, we can rewrite $\ln(x^{3})$ as $3\ln(x)$. So the equation $\ln(x^{3})-4\ln(x)=1$ becomes $3\ln(x)-4\ln(x)=1$.

Step2: Combine like terms

Combining the $\ln(x)$ terms on the left - hand side, we have $(3 - 4)\ln(x)=1$, which simplifies to $-\ln(x)=1$.

Step3: Isolate $\ln(x)$

Multiply both sides of the equation by $- 1$ to get $\ln(x)=-1$.

Step4: Convert to exponential form

Recall that if $\ln(x)=y$, then $x = e^{y}$. Since $\ln(x)=-1$, we have $x = e^{-1}=\frac{1}{e}$.

Answer:

$x=\frac{1}{e}$