Question

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

(a)${\log }_{4}x=3$ (b)${\log }_{10}0.01=x$

Short answer

1. The required value of x is64.
2. The required value of xis $-2$.

Step-by-step solution

Part a. Step 1. Given.

The given equation is ${\log }_{4}x=3$.

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 ${\log }_{4}x=3$with ${\log }_{a}c=b$we get $a=4,b=3,c=x$.

So, the equivalent exponential form is:

$a^b=c$

or, $4^3=x$

or, $x=64$

Hence, the required value of x is 64.

Part b. Step 1. Given.

The given equation is ${\log }_{10}0.01=x$.

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${\log }_{10}0.01=x$ with${\log }_{a}c=b$ we get $a=10,b=x,c=0.01$.

So, the equivalent exponential form is:

$a^b=c$

or, ${10}^x=0.01$

or,${10}^x=\frac{1}{100}$

or,${10}^x={10}^{-2}$

or, $x=-2$ [Since both bases are same, so the exponents must be same too]

Hence, the required value of x is $-2$.