Question

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

(a)${\log }_{x}1000=3$ (b)${\log }_{x}25=2$

Short answer

1. The required value of x is10.
2. The required value of xis 5.

Step-by-step solution

Part a. Step 1. Given.

The given equation is ${\log }_{x}1000=3$.

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

Comparing${\log }_{x}1000=3$ with${\log }_{a}c=b$ we get $a=x,b=3,c=1000$.

So, the equivalent exponential form is:

$a^b=c$

or, $x^3=1000$

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

or,$x=10$ [Equated the bases, since the exponents are same]

Hence, the required value of x is 10.

Part b. Step 1. Given.

The given equation is ${\log }_{x}25=2$.

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${\log }_{x}25=2$ with${\log }_{a}c=b$ we get $a=x,b=2,c=25$.

So, the equivalent exponential form is:

$a^b=c$

or,$x^2=25$

or,$x^2=5^2$

or,$x=5$ [Equated the bases, since the exponents are same]

Hence, the required value of x is 5.