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Chapter 3: Problem 5

Determine whether each \(x\) -value is a solution (or an approximate solution) of the equation. \(\log _{2}(x+3)=10\) (a) \(x=1021\) (b) \(x=17\) (c) \(x=10^{2}-3\)

Short Answer

Expert verified
For the logarithmic equation \( \log _{2} (x+3) = 10 \), only \( x = 1021 \) is a solution, while \( x = 17 \) and \( x = 10^{2}-3 \) are not.

Step by step solution

01

Understand the Logarithmic Equation

The logarithmic equation given is \( \log _{2}(x+3) = 10 \). Here, the base of the logarithm logarithm is 2, and the argument is (x + 3).
02

Substitute First Value of x

Substitute \(x = 1021\) into the equation, we have \( \log _{2}(1021+3) = 10 \). After adding you get \( \log _{2}(1024) \). The base 2 logarithm of 1024 is indeed 10, so this means that \( x = 1021\) is a solution.
03

Substitute Second Value of x

Substitute \(x = 17\) into the equation, we have \( \log _{2}(17+3) = 10 \). After adding you get \( \log _{2}(20) \). The base 2 logarithm of 20 is not 10, so \( x = 17 \) is not a solution.
04

Substitute Third Value of x

Substitute \(x = 10^{2}-3\) into the equation, we have \( \log _{2}((10^{2}-3)+3) = 10 \). After simplifying you get \( \log _{2}(100) \). The base 2 logarithm of 100 is not 10, so \(x = 10^{2}-3 \) is not a solution.

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Most popular questions from this chapter

Expanding a Logarithmic Expression In Exercises \(37-58\) , use the properties of logarithms to expand the expression as a sum, difference, and or constant multiple of logarithms. (Assume all variables are positive.) $$\ln \sqrt[3]{\frac{x}{y}}$$

Think About It For how many integers between 1 and 20 can you approximate natural logarithms, given the values \(\ln 2 \approx 0.6931,\) ln \(3 \approx 1.0986,\) and ln 5\(\approx 1.6094 ?\) Approximate these logarithms (do not use a calculator).

Using Properties of Logarithms In Exercises \(15-20\) , use the properties of logarithms to rewrite and simplify the logarithmic expression. $$\log _{4} 8$$

Expanding a Logarithmic Expression In Exercises \(37-58,\) use the properties of logarithms to expand the expression as a sum, difference, and or constant multiple of logarithms. (Assume all variables are positive.) $$\ln \sqrt{z}$$

Using Properties of Logarithms In Exercises \(59-66\) , approximate the logarithm using the properties of logarithms, given \(\log _{b} 2 \approx 0.3562, \log _{b} 3 \approx 0.5646,\) and \(\log _{b} 5 \approx 0.8271\) $$\log _{b} 10$$
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