Question

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

(a)$\ln x=3$ (b)$\ln e^2=x$

Short answer

1. The required value of x is$e^3$.
2. The required value of xis 2.

Step-by-step solution

Part a. Step 1. Given.

The given equation is $\ln x=3$.

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$\ln x=3$ with$\ln c=b$ we get $b=3,c=x$.

So, the equivalent exponential form is:

$e^b=c$

or, $e^3=x$

Hence, the required value of x is $e^3$.

Part b. Step 1. Given.

The given equation is $\ln e^2=x$.

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$\ln e^2=x$ with$\ln c=b$ we get $b=x,c=e^2$.

So, the equivalent exponential form is:

$e^b=c$

or,$e^x=e^2$

or,$x=2$ [Equated the exponents, since the bases are same]

Hence, the required value of x is 2.