Question

Logarithmic Equations Use the definition of the logarithmic function to find x.

(a)$\ln x=-1$ (b)$\ln \left(\frac{1}{e}\right)=x$

Short answer

1. The required value of x is$\frac{1}{e}$.
2. The required value of xis $-1$.

Step-by-step solution

Part a. Step 1. Given.

The given equation is $\ln x=-1$.

Part a. Step 2. To determine.

We have to find the value of x using the definition of the logarithmic function.

Part a. Step 3. Calculation.

We’ll use the definition of the logarithmic function $\ln c=b\Rightarrow e^b=c$.

Comparing$\ln x=-1$ with$\ln c=b$ we get $b=-1,c=x$.

So, the equivalent exponential form is:

$e^b=c$

or, $e^{-1}=x$

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

Hence, the required value of x is $\frac{1}{e}$.

Part b. Step 1. Given.

The given equation is $\ln \left(\frac{1}{e}\right)=x$.

Part b. Step 2. To determine.

We have to find the value of x using the definition of the logarithmic function.

Part b. Step 3. Calculation.

We’ll use the definition of the logarithmic function $\ln c=b\Rightarrow e^b=c$.

Comparing$\ln \left(\frac{1}{e}\right)=x$ with$\ln c=b$ we get $b=x,c=\frac{1}{e}$.

So, the equivalent exponential form is:

$e^b=c$

or,$e^x=\frac{1}{e}$

or,$e^x=e^{-1}$

or,$x=-1$ [Equated the exponents, since the bases are same]

Hence, the required value of x is $-1$.