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Chapter 4: Problem 11

Write each equation in its equivalent logarithmic form. $$2^{-4}=\frac{1}{16}$$

Short Answer

Expert verified
The logarithmic form of the given equation is \( \log_{2}\left(\frac{1}{16}\right) = -4 \).

Step by step solution

01

Identify the Base

Start by identifying the base in the exponential equation. The base is represented by the number being multiplied by itself, which in this case is \(2\). This is the base of the logarithmic function.
02

Identify the Exponent

The next step is to identify the exponent in the exponential equation. The exponent represents the number of times the base is multiplied by itself and is represented as \(-4\) in the equation. This will become the argument of the logarithm.
03

Identify the Result

Finally, the result of the exponential equation, in this case, \(\frac{1}{16}\), will be the characteristic of the logarithm.
04

Convert to Logarithmic Form

The logarithmic form of the equation is: \[ \log_{base}({result}) = exponent \]. Substituting the identified elements into this form, the equation becomes: \[ \log_{2}\left(\frac{1}{16}\right) = -4 \].

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Most popular questions from this chapter

In Exercises \(79-82,\) use a graphing utility and the change-of- base property to graph each function.$$ y=\log _{15} x $$

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ 2 \ln x-\frac{1}{2} \ln y $$

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ 7 \ln x-3 \ln y $$

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log \left(\frac{x}{1000}\right) $$

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ 2 \log _{4} x+3 \log _{b} y $$
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