Problem 143 - 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. If \(x=\frac{1}{k} \ln y,\) then \(y=e^{k x}\)

Step-by-Step Solution

Answer

The statement 'If \(x=\frac{1}{k} \ln y\), then \(y=e^{k x}\)' is True.

Step 1: Understanding Logarithm and Exponential Relationship

The base of the natural logarithm is the number 'e', and it has the property that \(\ln(e^x) = x\). Therefore, we can rewrite the initial statement \(x= \frac{1}{k}\ln y\) as \(x=\ln(y^{1/k})\). According to the properties of logarithms, we have \(y^{1/k}=e^x\).

Step 2: Substitution and Verification

Next, substitute the right side of the initial statement i.e., \(y=e^{k x}\) into our derived equation. This substitution gives us \(y=e^{k\ln(y^{1/k})}\). From the rule that \(\ln(a^b)=b\ln a\), we get \(y=e^{\ln(y)}\), which simplifies to \(y=y\).

Step 3: Conclusion

The initial statement is true since we end up with the relation \(y=y\) after our simplifications, which is always correct for all y. The statement that if \(x=\frac{1}{k} \ln y\), then \(y=e^{k x}\) is indeed true.