Question

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

(a)${\log }_2\left(\frac{1}{2}\right)=x$ (b)${\log }_{10}x=-3$

Short answer

1. The required value of x is$-1$.
2. The required value of xis 0.001.

Step-by-step solution

Part a. Step 1. Given.

The given equation is ${\log }_2\left(\frac{1}{2}\right)=x$.

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${\log }_2\left(\frac{1}{2}\right)=x$ with${\log }_{a}c=b$ we get $a=2,b=x,c=\frac{1}{2}$.

So, the equivalent exponential form is:

$a^b=c$

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

or,$2^x=2^{-1}$

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

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

Part b. Step 1. Given.

The given equation is ${\log }_{10}x=-3$.

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${\log }_{10}x=-3$ with${\log }_{a}c=b$ we get $a=10,b=-3,c=x$.

So, the equivalent exponential form is:

$a^b=c$

or,${10}^{-3}=x$

or,$\frac{1}{1000}=x$

or,$0.001=x$

Hence, the required value of x is 0.001.