Problem 142 - 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. \(\log _{b} x\) is the exponent to which \(b\) must be raised to obtain \(x\).

Step-by-Step Solution

Answer

The statement is true.

Step 1: Understanding the Definition of Logarithms

Recognize that the fundamental definition of a logarithm is: \[ \log _{b} x \] is the exponent \(y\) that we need, to raise the base \(b\) to get the number \(x\). In other words, if \( b^y = x \), then \( \log _{b} x = y \). This definition is necessary to proceed with the problem.

Step 2: Verify the Statement

Check the given statement \'\(\log _{b} x \) is the exponent to which \(b\) must be raised to obtain \(x\).\' Against the definition of a logarithm, you can see that it directly aligns with the definition established in step 1.

Step 3: Conclusion

Since the given statement aligns perfectly with the definition of a logarithm, the statement is therefore correct or 'True'. No changes are needed.