Question

Convert to logarithmic form:

$$\begin{gathered} ⓐ4^3=64 \\ ⓑ4^{\frac{1}{3}}=\sqrt[3]{4} \\ ⓒ{\left(\frac{1}{2}\right)}^x=\frac{1}{32} \end{gathered}$$

Short answer

The logarithmic forms for the given expressions are,

$$\begin{gathered} a.3={\log }_4\left(64\right) \\ b.\frac{1}{3}={\log }_4\left(\sqrt[3]{4}\right) \\ c.x={\log }_{\frac{1}{2}}\left(\frac{1}{32}\right) \end{gathered}$$

Step-by-step solution

Step 1 Given expressions are,

$$\begin{gathered} ⓐ4^3=64 \\ ⓑ4^{\frac{1}{3}}=\sqrt[3]{4} \\ ⓒ{\left(\frac{1}{2}\right)}^x=\frac{1}{32} \end{gathered}$$

The objective is, we need to find the logarithmic form for the given expressions.

Part a Step 2 src="data:image/svg+xml;base64,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" localid="1645433180090" src="https://studysmarter-mediafiles.s3.amazonaws.com/media/textbook-exercise-images/e7fb8d7b-62a5-4376-8de7-70539528bdb1.svg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA4OLDUDE42UZHAIET%2F20221108%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20221108T134537Z&X-Amz-Expires=90000&X-Amz-SignedHeaders=host&X-Amz-Signature=53a5c625c53acf5b02deea14c1bc55724c8b7a4da3669632d017f80830e7be47" src="https://studysmarter-mediafiles.s3.amazonaws.com/media/textbook-exercise-images/e7fb8d7b-62a5-4376-8de7-70539528bdb1.svg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA4OLDUDE42UZHAIET%2F20220223%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20220223T161103Z&X-Amz-Expires=90000&X-Amz-SignedHeaders=host&X-Amz-Signature=e76258445c77bdc45e21ebd928ce7586ea72ca007f2a1ad9945fd66604625b21" 43 = 64 .

$$\begin{gathered} 4^3=64 \\ y={\log }_a\left(x\right) \\ 3={\log }_4\left(64\right) \end{gathered}$$

Part b Step 2    413=43

$$\begin{gathered} 4^{\frac{1}{3}}=\sqrt[3]{4} \\ y={\log }_a\left(x\right) \\ \frac{1}{3}={\log }_4\left(\sqrt[3]{4}\right) \end{gathered}$$

So, the answer is, $\frac{1}{3}={\log }_4\left(\sqrt[3]{4}\right)$.

Part c Step 2 12x=132.

$$\begin{gathered} {\left(\frac{1}{2}\right)}^x=\frac{1}{32} \\ y={\log }_a\left(x\right) \\ x={\log }_{\frac{1}{2}}\left(\frac{1}{32}\right) \end{gathered}$$

So, the answer is, $x={\log }_{\frac{1}{2}}\left(\frac{1}{32}\right)$.