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Chapter 3: Problem 10

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
Of the values given for \(x\), only \(x = 1021\) is a solution to the equation \(\log _{2}(x+3)=10\).

Step by step solution

01

Transforming the Equation

First, transform the logarithmic equation into an exponential one. This can be done using the relationship between logarithmic and exponential functions, \(a = log_b(c) <=> b^a = c\). So the equation \(\log _{2}(x+3)=10\) can be rewritten as \(2^{10} = x + 3\).
02

Calculate the Value of x

Solving \(2^{10} = x + 3\), you get \(x = 2^{10} - 3\).
03

Checking for Solution (a)

Substitute \(x = 1021\) into the equation \(2^{10} = x + 3\), you get \(2^{10} = 1024\), so \(x = 1021\) is a true solution.
04

Checking for Solution (b)

Substitute \(x = 17\) into the equation \(2^{10} = x + 3\), you get \(2^{10} = 20\), so \(x = 17\) is not a solution.
05

Checking for Solution (c)

Substitute \(x = 10^{2} - 3\) into the equation \(2^{10} = x + 3\), you get \(2^{10} = 100\), so \(x = 10^{2} - 3\) is not a solution.

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Most popular questions from this chapter

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