Question

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

(a)${\log }_7\left(\frac{1}{49}\right)=x$ (b)${\log }_{2}x=5$

Short answer

1. The required value of x is$-2$.
2. The required value of xis 32.

Step-by-step solution

Part a. Step 1. Given.

The given equation is ${\log }_7\left(\frac{1}{49}\right)=x$.

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${\log }_7\left(\frac{1}{49}\right)=x$ with${\log }_{a}c=b$ we get $a=7,b=x,c=\frac{1}{49}$.

So, the equivalent exponential form is:

$a^b=c$

or, $7^x=\frac{1}{49}$

or,$7^x=7^{-2}$

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

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

Part b. Step 1. Given.

The given equation is ${\log }_{2}x=5$.

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${\log }_{2}x=5$ with${\log }_{a}c=b$ we get $a=2,b=5,c=x$.

So, the equivalent exponential form is:

$a^b=c$

or,$2^5=x$

or,$32=x$

Hence, the required value of x is 32.