Problem 102 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the exact value of the logarithm without using a calculator. If this is not possible, state the reason.$$\ln \sqrt[5]{e^{3}}$$.

Step-by-Step Solution

Answer

The exact value of the logarithm is \(\frac{3}{5}\).

Step 1: Convert the 5th root into exponential form

The 5th root of a number is the same as raising that number to the power of \(\frac{1}{5}\). Therefore, \(\sqrt[5]{e^{3}} = (e^{3})^{\frac{1}{5}}\).

Step 2: Apply the rule of exponents

Using the rule that \(a^{mn} = (a^{m})^{n}\), It can be written as \(e^{3*\frac{1}{5}} = e^{\frac{3}{5}}\).

Step 3: Apply the rule of logarithms

The natural logarithm \(\ln e^{\frac{3}{5}}\) can be simplified using the rule of logarithms that \(\ln a^n = n \ln a\). In this case, \(\ln a^n = n\). Because the natural logarithm base \(e\) of \(e\) is 1, this simplifies to \(\frac{3}{5} * 1 = \frac{3}{5}\).

Key Concepts

Exponential Form

Understanding the exponential form is essential for simplifying complex mathematical problems. It involves expressing numbers as a base raised to a specific power. In the context of the exercise given, the exponent plays a crucial role in finding the exact value of a logarithm.

For example, when dealing with roots, any root can be rewritten in exponential form. The fifth root of a number, as seen in the problem, \(\sqrt[5]{e^{3}}\), is rewritten as \(e^{3})^{\frac{1}{5}}\). This conversion is integral when trying to simplify expressions before applying logarithms. Exponents allow us to condense multiplication of the same base and represent repeated multiplication compactly. Grasping this concept paves the way for manipulating logarithmic expressions efficiently.

Rule of Exponents

One of the fundamental rules of exponents, crucial for solving the given problem, is that \(a^{mn} = (a^{m})^{n}\). This rule supports the simplification of expressions where an exponent is raised to another exponent.

By applying this rule, one can reduce the expression \(e^{3})^{\frac{1}{5}}\) to \(e^{\frac{3}{5}}\). Here, the base \(e\) remains constant while the exponents are multiplied. This property is particularly useful in conjunction with logarithms because it allows us to extract exponents out of logarithms, transforming multiplicative processes into additive ones. Mastery of this rule is indispensable when working with exponents and their corresponding logarithmic forms.

Logarithm Properties

Logarithm properties are indispensable tools in algebra that facilitate the simplification and solving of logarithmic equations. Specifically, the property \(\ln a^n = n \ln a\) allows you to move the exponent in a logarithmic function to the front, making the equation more manageable.

For our exercise, \(\ln e^{\frac{3}{5}}\) can be simplified to \(\frac{3}{5} \ln e\), and given that the natural logarithm of \(e\) is 1 (\(\ln e = 1\)), this further simplifies to \(\frac{3}{5} \). This highlights how logarithm properties are instrumental in solving expressions involving natural logarithms. It's vital to have a firm understanding of these properties to convert complex logarithmic expressions into simpler ones and identify solutions without the aid of a calculator.