Problem 26 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the definition of logarithmic function to evaluate the function at the indicated value of \(x\) without using a calculator. (Value) $$x=16$$ $$x=\frac{1}{4}$$ $$x=\frac{1}{1000}$$ $$x=100,000$$ (Function) $$g(x)=\log _{10} x$$

Step-by-Step Solution

Answer

The values of the function are undeterminable for \( x = 16 \), \( -2 \) for \( x = 1/4 \), \( -3 \) for \( x = 1/1000 \), and \( 5 \) for \( x = 100,000 \).

Step 1: Evaluate for \( x = 16 \)

Put this value \( x = 16 \) into the function \( g(x) = \log_{10} x \). We only have whole number solutions when \( x \) equals to a power of 10. Since 16 is not a power of 10, we cannot determine a whole number solution for this case. It's only calculable with a calculator.

Step 2: Evaluate for \( x = 1/4 \)

Put this value \( x = 1/4 \) into the function \( g(x) = \log_{10} x \). Since \( 1/4 = 10^{-2} \) , we have \( g(x) = \log_{10}(10^{-2}) = -2 \)

Step 3: Evaluate for \( x = 1/1000 \)

Put this value \( x = 1/1000 \) into the function \( g(x) = \log_{10} x \). Since \( 1/1000 = 10^{-3} \) , we have \( g(x) = \log_{10}(10^{-3}) = -3 \)

Step 4: Evaluate for \( x = 100,000 \)

Put this value \( x = 100,000 \) into the function \( g(x) = \log_{10} x \). Since \( 100,000 = 10^{5} \) , we have \( g(x) = \log_{10}(10^{5}) = 5 \)