Question

Solve:

$$\begin{gathered} a){\log }_a\left(121\right)=2 \\ b)\ln \left(x\right)=7 \end{gathered}$$

Short answer

Part (a) The value of $a$is$11$.

Part (b) The value of$x$is$e^7$.

Step-by-step solution

Step 1. Given information.

The given expressions are,

$$\begin{gathered} a){\log }_a\left(121\right)=2 \\ b)\ln \left(x\right)=7 \end{gathered}$$

Step 2. Solving part (a).

Here, we will convert the given expression to the exponential form and solve the quadratic equation.

$$\begin{gathered} {\log }_a\left(121\right)=2 \\ {a}^2=121 \\ a=11,-11 \end{gathered}$$

The base of a logarithmic function cannot be negative hence the value of $a$is $11$.

Checking the result.

$$\begin{gathered} {\log }_a\left(121\right)=2 \\ {\log }_{11}\left(121\right)=2 \\ {\left(11\right)}^2=121 \\ 121=121 \end{gathered}$$

Therefore, this is true.

Step 3. Solving part (b).

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

$$\begin{gathered} \ln \left(x\right)=7 \\ {e}^7=x \end{gathered}$$

Checking the result.

$$\begin{gathered} \ln \left(x\right)=7 \\ \ln \left(e^7\right)=7 \\ {e}^7=e^7 \end{gathered}$$

Therefore, this is true.