Question

rewrite the following in exponential form: a) \\(\log_{4}\left(\frac{1}{4}\
ight) = -1\\) b) \\(\log_{10}(127) = x\\) c) \\(\log_{e}(x) = 0\\)

Explanation:

Part (a)

Step1: Recall log to exponential rule

The logarithmic equation \(\log_{b}(y) = k\) can be rewritten in exponential form as \(b^{k}=y\). Here, \(b = 4\), \(y=\frac{1}{4}\), and \(k=- 1\).

Step2: Apply the rule

Substitute the values into the exponential form formula. So we get \(4^{-1}=\frac{1}{4}\).

Step1: Recall log to exponential rule

For a logarithmic equation \(\log_{b}(y)=k\), the exponential form is \(b^{k} = y\). Here, \(b = 10\), \(y = 127\), and \(k=x\).

Step2: Apply the rule

Substitute the values into the formula. We obtain \(10^{x}=127\).

Step1: Recall log to exponential rule

The formula to convert \(\log_{b}(y)=k\) to exponential form is \(b^{k}=y\). Here, \(b = e\), \(y = x\), and \(k = 0\).

Step2: Apply the rule

Substitute the values into the formula. We have \(e^{0}=x\).

Answer:

\(4^{-1}=\frac{1}{4}\)

Part (b)