Question

Solve:

$$\begin{gathered} a){\log }_3\left(4x+3\right)=3 \\ b)\ln \left(e^{4x}\right)=4 \end{gathered}$$

Short answer

Part (a) The value of $x$is $6$.

Part (b) The value of$x$is$1$.

Step-by-step solution

Step 1. Given information.

The given expressions are,

$$\begin{gathered} a){\log }_3\left(4x+3\right)=3 \\ b)\ln \left(e^{4x}\right)=4 \end{gathered}$$

Step 2. Solving part (a).

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

$$\begin{gathered} {\log }_3\left(4x+3\right)=3 \\ 4x+3=3^3 \\ 4x+3=27 \\ 4x=24 \\ x=6 \end{gathered}$$

Checking the result.

$$\begin{gathered} {\log }_3\left(4\left(6\right)+3\right)=3 \\ 4\left(6\right)+3=3^3 \\ 24+3=27 \\ 27=27 \end{gathered}$$

Therefore, this is true.

Step 3. Solving part (b).

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

$$\begin{gathered} \ln \left(e^{4x}\right)=4 \\ {e}^4=e^{4x} \\ 4x=4 \\ x=1 \end{gathered}$$

Checking the result.

$$\begin{gathered} \ln \left(e^{4\left(1\right)}\right)=4 \\ {e}^4=e^4 \end{gathered}$$

Therefore, this is true.