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Chapter 4: Problem 1

Write each equation in its equivalent exponential form. $$4=\log _{2} 16$$

Short Answer

Expert verified
The equivalent exponential form of the given logarithmic equation \(4 = \log_2 16\) is \(2^4 = 16\).

Step by step solution

01

Identify the base, exponent, and result in the logarithmic equation

In the given logarithmic equation \(\log_2 16 = 4\), the base of the logarithm is 2, the result of the logarithm (the number that the logarithm is equal to) is 4, and the number that the logarithm is acting on (inside the logarithm) is 16.
02

Apply the relationship between logarithms and exponents

A logarithmic equation \(\log_b a = c\) can be converted into the exponential equation \(b^c = a\). Here, \(b\) is the base, \(a\) is the number inside the logarithm, and \(c\) is the result. For the given equation \(\log_2 16 = 4\), this gives \(2^4 = 16\).
03

Check the solution

After transforming the logarithmic equation into an exponential one, it can be checked by ensuring that the equation holds true. For \(2^4 = 16\), raising 2 to the power of 4 indeed gives 16, so the exponential form is correct.

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Most popular questions from this chapter

Graph \(y=\log x, y=\log (10 x),\) and \(y=\log (0.1 x)\) in the same viewing rectangle. Describe the relationship among the three graphs. What logarithmic property accounts for this relationship?

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. $$ \log N^{-6} $$

Solve each equation in Exercises \(89-91 .\) Check each proposed solution by direct substitution or with a graphing utility. $$\ln (\ln x)=0$$

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. $$ \log _{7}(7 x) $$

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. $$ \log _{4}\left(\frac{64}{y}\right) $$
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