The image shows two tasks that ask you to rewrite equations in different forms: logarithmic and exponential.

Part (a):

Rewrite $5^{-2} = \frac{1}{25}$ as a logarithmic equation.

This equation can be rewritten in logarithmic form using the general rule: $a^x = b \quad \text{is equivalent to} \quad \log_a(b) = x.$

In this case:

• The base is $5$,
• The result is $\frac{1}{25}$,
• The exponent is $-2$.

Thus, the logarithmic form is: $\log_5\left(\frac{1}{25}\right) = -2.$

Part (b):

Rewrite $\log_2 16 = 4$ as an exponential equation.

This logarithmic equation can be rewritten in exponential form using the rule: $\log_a(b) = x \quad \text{is equivalent to} \quad a^x = b.$

Here:

• The base is $2$,
• The exponent is $4$,
• The result is $16$.

Thus, the exponential form is: $2^4 = 16.$

Would you like any further details on logarithms or exponential equations?

Here are 5 related questions you could explore:

1. How do you convert any logarithmic equation to exponential form?
2. Why is $5^{-2} = \frac{1}{25}$ true based on exponent rules?
3. Can you have a negative base in logarithmic or exponential equations?
4. What are the real-world applications of logarithmic equations?
5. How can you solve an exponential equation like $5^x = 25$ without converting to logarithmic form?

Tip: A quick way to verify exponential equations is to calculate the power manually. For example, $2^4 = 2 \times 2 \times 2 \times 2 = 16$.