Question

Solve:

$$\begin{gathered} a){\log }_2\left(5x-1\right)=6 \\ b)\ln \left(e^{3x}\right)=6 \end{gathered}$$

Short answer

Part (a) The value of $x$is $13$.

Part (b) The value of$x$is$2$.

Step-by-step solution

Step 1. Given information.

The given expressions are,

$$\begin{gathered} a){\log }_2\left(5x-1\right)=6 \\ b)\ln \left(e^{3x}\right)=6 \end{gathered}$$

Step 2. Solving part (a).

Now, we will convert the given expression to exponential form and simplify.

$$\begin{gathered} {\log }_2\left(5x-1\right)=6 \\ 5x-1=2^6 \\ 5x-1=64 \\ 5x=65 \\ x=13 \end{gathered}$$

Checking the result.

$$\begin{gathered} {\log }_2\left(5\left(13\right)-1\right)=6 \\ 5\left(13\right)-1=2^6 \\ 65-1=64 \\ 64=64 \end{gathered}$$

Therefore, this is true.

Step 3. Solving part (b).

Now, we will convert the given expression to exponential form and simplify.

$$\begin{gathered} \ln \left(e^{3x}\right)=6 \\ {e}^6=e^{3x} \\ 3x=6 \\ x=2 \end{gathered}$$

Checking the result.

$$\begin{gathered} \ln \left(e^{3\left(2\right)}\right)=6 \\ {e}^6=e^{3\left(2\right)} \\ {e}^6=e^6 \end{gathered}$$

Therefore, this is true.