Question

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

(a)${\log }_{4}2=x$ (b)${\log }_{4}x=2$

Short answer

1. The required value of x is$\frac{1}{2}$.
2. The required value of xis 16.

Step-by-step solution

Part a. Step 1. Given.

The given equation is ${\log }_{4}2=x$.

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${\log }_{4}2=x$ with${\log }_{a}c=b$ we get $a=4,b=x,c=2$.

So, the equivalent exponential form is:

$a^b=c$

or, $4^x=2$

or,$4^x=4^{\frac{1}{2}}$

or,$x=\frac{1}{2}$ [Equated the exponents, since the bases are same]

Hence, the required value of x is $\frac{1}{2}$.

Part b. Step 1. Given.

The given equation is ${\log }_{4}x=2$.

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${\log }_{4}x=2$ with${\log }_{a}c=b$ we get $a=4,b=2,c=x$.

So, the equivalent exponential form is:

$a^b=c$

or,$4^2=x$

or,$16=x$

Hence, the required value of x is 16.