Problem 143 - Algebra and Trigonometry

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$ \text { If } x=\frac{1}{k} \ln y, \text { then } y=e^{k x} $$

Step-by-Step Solution

Answer

The statement 'If \(x=\frac{1}{k} \ln y\), then \(y=e^{kx}\)' is indeed true. The right-side expression does equal the left-side expression after simplification.

Step 1: Identifying the Pursuit

Identify the equation given and understand that the task is to ascertain the validity of the equation by applying the rules of logarithms.

Step 2: Examine the Right-Side expression

Consider the right hand side of the equation, which is \(y=e^{kx}\). You could rewrite \(e^{kx}\) as \(e^x\) raised to the power of \(k\). So, the equation becomes \(y=(e^x)^k\).

Step 3: Relating to the Left-Side expression

Relate the simplified equation of the right-side expression to the left-side expression. Since the base of the natural logarithm is \(e\), equation \(x=\frac{1}{k} \ln y\) is equivalent to equation \(y=(e^x)^k\). Thus, the given statement is true.

Key Concepts

Natural Logarithm

The natural logarithm, often denoted as \(ln\), is a logarithm with the base \(e\). The number \(e\) is a mathematical constant approximately equal to 2.71828. Unlike common logarithms that use base 10, the natural logarithm is widely used in mathematics because of its simplicity in calculus.

• When you see \(ln y\), it means you're finding the power to which \(e\) must be raised to get \(y\).
• This is closely related to exponential functions. If \(ln y = x\), then it follows that \(y = e^x\).

Understanding natural logarithms is crucial for solving equations that involve exponential growth or decay, such as the problems you may encounter in calculus or differential equations.
Knowing how to manipulate \(ln\) helps in rearranging logarithmic equations effectively.

Exponential Functions

Exponential functions are a class of mathematical functions where the variable appears in the exponent. The most common form you will encounter is \(e^x\), where \(e\) is Euler's number.

• An important property of exponential functions is their continuous growth or decay rate, which is constant.
• The function \(e^{kx}\) represents exponential growth or decay depending on the sign of \(k\).

This makes exponential functions invaluable in modeling real-world phenomena such as population growth, radioactive decay, and compound interest.
It is crucial to see the relationship between an exponential function and its corresponding logarithmic form. These relationships simplify solving complex logarithmic equations.

Equation Validation

Equation validation is the process of confirming whether a mathematical statement or equation is true. For the equation in the exercise \(x = \frac{1}{k} \ln y\), you need to connect this to the equation \(y = e^{kx}\).

• Start by isolating \(y\) from the first equation: \(\ln y = kx\), which becomes \(y = e^{kx}\).
• This process involves manipulating equations and confirming that both sides can represent the same relationship.

By understanding both sides of the equation thoroughly, you ensure that the statement is accurate or identify necessary corrections if it is false.
This is a crucial skill for checking your work in mathematics and ensuring that calculations are grounded in correct principles.

Properties of Logarithms

Logarithms have several useful properties that simplify complex mathematical problems. These properties help in transforming and solving logarithmic equations.

• Product Rule: \(\ln(ab) = \ln a + \ln b\)
• Quotient Rule: \(\ln\left(\frac{a}{b}\right) = \ln a - \ln b\)
• Power Rule: \(\ln(a^b) = b\ln a\)
• Inverse Property: \(e^{\ln x} = x\) and \(\ln(e^x) = x\)

Applying these properties allows for the conversion of complex logarithmic expressions into simpler forms.
Using these rules, you can manipulate logarithmic equations like the one given, turning them into exponential form and validate the provided statements.