Question

Write the logarithm in terms of natural logarithms. \(\log _{5} 8\)

Short answer

The base 5 logarithm \(\log _{5} 8\) can be expressed in terms of natural logarithms as \(\frac{\ln 8}{\ln 5}\).

Step-by-step solution

Identify the parts of the logarithm

In the given expression \(\log _{5} 8\), '5' is the base of the logarithm and '8' is the argument of the logarithm.

Write down the change of base formula

The change of base formula is \(\log_b a = \frac{\ln a}{\ln b}\). Here, 'b' is the original base (5 in this case) and 'a' is the argument of the logarithm (8 in this case). 'ln' stands for natural logarithm.

Substitute the values into the change of base formula and simplify

Substituting '5' for 'b' and '8' for 'a' into the formula gives \(\log _{5} 8 = \frac{\ln 8}{\ln 5}\). This fraction represents the original logarithm expressed in terms of natural logarithms.

Key concepts

Understanding Natural Logarithms

Natural logarithms are a special class of logarithms with a base of 'e', where 'e' is an irrational and transcendental number approximately equal to 2.71828. Historically, natural logarithms were first described in the context of compound interest rates, where 'e' arises naturally as the limit of \( (1 + 1/n)^n \) as 'n' approaches infinity.

In mathematics, particularly calculus, natural logarithms are indispensable due to their unique properties, such as the derivative of the natural logarithm function, \( \ln(x) \), being \( 1/x \). They also play a crucial role in solving equations involving exponential growth and decay. Students often come across the notation \( \ln(x) \) as a shorthand for \( \log_e(x) \) in their studies, as the base 'e' logarithm is so common that it assumes its own symbol.

Logarithm Properties

Logarithms possess specific properties that make them powerful tools for solving and simplifying mathematical expressions. Here are a few key properties:

• Product Property: The logarithm of a product is equal to the sum of the logarithms of the factors, expressed as \( \log_b(mn) = \log_b(m) + \log_b(n) \).
• Quotient Property: The logarithm of a quotient is the difference of the logarithms, represented as \( \log_b(m/n) = \log_b(m) - \log_b(n) \).
• Power Property: The logarithm of a power allows the exponent to be brought to the front, exemplified by \( \log_b(m^n) = n \cdot \log_b(m) \).
• Change of Base: Logarithms can be converted from one base to another using the formula \( \log_b(a) = \frac{\ln a}{\ln b} \), a process important for computational purposes when using calculators or when a specific base, like 'e', is more convenient.

These properties enable mathematicians and students alike to manipulate logarithmic expressions into forms that are easier to work with or solve.

Manipulating Logarithmic Expressions

When you're faced with complex logarithmic expressions, understanding how to manipulate them using the properties of logarithms is the key to simplifying or solving them. For example, to express \( \log _{5} 8 \) in terms of natural logarithms, we use the change of base formula mentioned in the exercise solution.

By applying the change of base formula, we convert \( \log _{5} 8 \) into a ratio of natural logarithms \( \frac{\ln 8}{\ln 5} \) because natural logarithms are involved. Simplifying further may involve using logarithm properties, or for numerical answers, computation using a calculator. The simplicity of natural logarithms, along with their innate relationship to the number 'e', often makes them the preferred choice when dealing with exponential and logarithmic equations.

If the original expression had been an equation, you might need to exponentiate both sides or utilize logarithm properties to isolate the variable involved. This kind of expression manipulation requires practice to master, but once understood, it opens the door to solving a wide range of mathematical problems involving logarithms.