Problem 141 - College Algebra Essentials

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\text { The domain of } f(x)=\log _{2} x \text { is }(-\infty, \infty)$$.

Step-by-Step Solution

Answer

False. The correct statement is: If \(\log (x+3)=2\), then \(10^{2}=x+3\).

Step 1: Interpret the given logarithmic equation

The logarithm equation given is \(\log (x+3)=2\). This implies that 10 raised to the power of 2 equals x+3, since the base of the logarithm, if not mentioned, is 10. So we get \(10^2=x+3\), which further simplifies to \(x+3=100\).

Step 2: Solve for x in the logarithmic equation

Solving the above equation for x, we get \(x=100-3\) which simplifies to \(x=97\).

Step 3: Analyze the proposed exponential equation

The proposed exponential equation in the statement is \(e^{2}=x+3\). Here the base of the exponential function is e (approximately 2.718), which is not the same as the base of the logarithm in the given equation. So, without any calculations, this equation cannot be true given the original logarithmic equation.

Step 4: Propose a correction to make the statement true

To make the statement true, the base of the original logarithmic equation (which is 10) should be applied to the exponential function. So the corrected statement would be: If \(\log (x+3)=2\), then \(10^{2}=x+3\).