Question

Logarithmic Form Express the equation in logarithmic form.

(a)${10}^4=10,000$ (b)$5^{-2}=\frac{1}{25}$

Short answer

1. Logarithmic form of ${10}^4=10,000$is ${\log }_{10}(10,000)=4$.
2. Logarithmic form of $5^{-2}=\frac{1}{25}$is width="110">log5125=−2.

Step-by-step solution

Part a. Step 1. Given.

The given exponential form is ${10}^4=10,000$.

Part a. Step 2. To determine.

We have to find the exponential form.

Part a. Step 3. Calculation.

We’ll use the formula: $a^b=c$is equivalent to ${\log }_{a}c=b$.

Comparing we find the values of a, b, c. Then plug them in the exponential form.

Exponential form ($a^b=c$)	a	b	c	Logarithmic form (${\log }_{a}c=b$)
${10}^4=10,000$	10	4	10,000	${\log }_{10}(10,000)=4$

Hence, logarithmic form of ${10}^4=10,000$is ${\log }_{10}(10,000)=4$.

Part b. Step 1. Given.

The given exponential form is $5^{-2}=\frac{1}{25}$.

Part b. Step 2. To determine.

We have to find the exponential form.

Part b. Step 3. Calculation.

We’ll use the formula:$a^b=c$ is equivalent to ${\log }_{a}c=b$.

Comparing we find the values of a, b, c. Then plug them in the exponential form.

Exponential form ($a^b=c$)	a	b	c	Logarithmic form (${\log }_{a}c=b$)
$5^{-2}=\frac{1}{25}$	5	$-2$	$\frac{1}{25}$	${\log }_5\left(\frac{1}{25}\right)=-2$

Hence, logarithmic form of $5^{-2}=\frac{1}{25}$is ${\log }_5\left(\frac{1}{25}\right)=-2$.