Question

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

(a)${\log }_{x}6=\frac{1}{2}$ (b)${\log }_{x}3=\frac{1}{3}$

Short answer

1. The required value of x is36.
2. The required value of xis 27.

Step-by-step solution

Part a. Step 1. Given.

The given equation is ${\log }_{x}6=\frac{1}{2}$.

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 ${\log }_{a}c=b\Rightarrow a^b=c$.

Comparing${\log }_{x}6=\frac{1}{2}$ with${\log }_{a}c=b$ we get $a=x,b=\frac{1}{2},c=6$.

So, the equivalent exponential form is:

$a^b=c$

or, $x^{\frac{1}{2}}=6$

or, ${\left(x^{\frac{1}{2}}\right)}^2={\left(6\right)}^2$ [Square both sides]

or,$x=36$ [Equated the bases, since the exponents are same]

Hence, the required value of x is 36.

Part b. Step 1. Given.

The given equation is ${\log }_{x}3=\frac{1}{3}$.

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 ${\log }_{a}c=b\Rightarrow a^b=c$.

Comparing${\log }_{x}3=\frac{1}{3}$ with${\log }_{a}c=b$ we get $a=x,b=\frac{1}{3},c=3$.

So, the equivalent exponential form is:

$a^b=c$

or,$x^{\frac{1}{3}}=3$

or, ${\left(x^{\frac{1}{3}}\right)}^3=3^3$[Cube both sides]

or, $x=27$

Hence, the required value of x is 27.