Question

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

(a)${\log }_{3}x=-2$ (b)${\log }_{5}125=x$

Short answer

1. The required value of x is$\frac{1}{9}$.
2. The required value of xis 3.

Step-by-step solution

Part a. Step 1. Given.

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

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

So, the equivalent exponential form is:

$a^b=c$

or, $3^{-2}=x$

or,$\frac{1}{9}=x$

Hence, the required value of x is $\frac{1}{9}$.

Part b. Step 1. Given.

The given equation is ${\log }_{5}125=x$.

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${\log }_{5}125=x$ with${\log }_{a}c=b$ we get $a=5,b=x,c=125$.

So, the equivalent exponential form is:

$a^b=c$

or, $5^x=125$

or,$5^x=5^3$

or, $x=3$ [Since both bases are same, so the exponents must be same too]

Hence, the required value of x is 3.