Question

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

(a)${\log }_{x}16=4$ (b)${\log }_{x}8=\frac{3}{2}$

Short answer

1. The required value of x is2.
2. The required value of xis 4.

Step-by-step solution

Part a. Step 1. Given.

The given equation is ${\log }_{x}16=4$.

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${\log }_{x}16=4$ with${\log }_{a}c=b$ we get $a=x,b=4,c=16$.

So, the equivalent exponential form is:

$a^b=c$

or, $x^4=16$

or,$x^4=2^4$

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

Hence, the required value of x is 2.

Part b. Step 1. Given .

The given equation is ${\log }_{x}8=\frac{3}{2}$.

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${\log }_{x}8=\frac{3}{2}$ with${\log }_{a}c=b$ we get $a=x,b=\frac{3}{2},c=8$.

So, the equivalent exponential form is:

$a^b=c$

or,$x^{\frac{3}{2}}=8$

or,${\left(x^{\frac{1}{2}}\right)}^3=2^3$ [Using exponent of exponent property]

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

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

or,$x=4$

Hence, the required value of x is 4.