Question

Solve the logarithmic equation algebraically. Approximate the result to three decimal places.$2-6\ln x=10$

Short answer

The approximate solution to the equation $2-6\ln x=10$ is $x\approx 0.263$.

Step-by-step solution

Isolate the Natural Logarithm

Start by isolating the log term by subtracting $2$ from each side of the equation, we get: $-6\ln x=10-2$, which simplifies to $-6\ln x=8$.

Remove the coefficient from the natural logarithm

Divide both sides by $-6$ to remove the coefficient from the natural logarithm, thus getting: $\ln x=-\frac{4}{3}$.

Remove the logarithm

To remove the log from the equation and isolate $x$, use the property that says if $\ln a=b$, then $a=e^b$. Here, $e^{-\frac{4}{3}}=x$.

Approximate

Now, compute $e^{-\frac{4}{3}}$ using a calculator and round to three decimal places. After this step, we get $x\approx 0.263$.