Evaluate the logarithm log6 1/36 – In the realm of mathematics, the concept of logarithms plays a pivotal role. One particular logarithm that we will delve into is log6 1/36. In this comprehensive guide, we will embark on a journey to evaluate this logarithm, exploring its properties and uncovering its applications.

To begin our exploration, we will establish a solid understanding of logarithmic functions, their characteristics, and how to solve logarithmic equations. With this foundation, we will then delve into the intricacies of evaluating logarithms, including the change-of-base formula.

Logarithmic Functions

Logarithmic functions are mathematical functions that are the inverse of exponential functions. They are used to find the exponent to which a base must be raised to produce a given number.

The logarithm of a number to the base bis the exponent to which bmust be raised to produce that number. It is written as log_b(x) , where xis the number.

Properties of Logarithms

• The logarithm of a product is equal to the sum of the logarithms of the factors.
• The logarithm of a quotient is equal to the difference of the logarithms of the numerator and denominator.
• The logarithm of a power is equal to the exponent multiplied by the logarithm of the base.
• The logarithm of 1 is 0.
• The logarithm of the base is 1.

Examples of Logarithmic Equations

• log₂(8) = 3
• log₁₀(100) = 2
• log_e(e ^x) = x

Evaluating Logarithms

Evaluating logarithms involves finding the exponent to which a base must be raised to produce a given number. This process is used extensively in various mathematical and scientific applications.

Change-of-Base Formula

The change-of-base formula allows us to convert a logarithm with an arbitrary base to a logarithm with a different base. The formula is:

log_ba = log _ca / log _cb

where ais the number, bis the new base, and cis the original base.

Examples of Evaluating Logarithms

Example 1:Evaluate log ₆1/36

• Using the change-of-base formula with c=10:

log₆1/36 = log ₁₀1/36 / log ₁₀6

• Evaluating the logarithms:

log₁₀1/36 = -1.5563 log ₁₀6 = 0.7782

• Substituting the values:

log₆1/36 = -1.5563 / 0.7782 = -2

Example 2:Evaluate log ₁₂81

• Using the change-of-base formula with c=3:

log₁₂81 = log ₃81 / log ₃12

• Evaluating the logarithms:

log₃81 = 4 log ₃12 = 2.2619

• Substituting the values:

log₁₂81 = 4 / 2.2619 = 1.7687

Evaluating log6 1/36

To evaluate log6 1/36, we can use the properties of logarithms to convert it to an equivalent expression with a more convenient base. We can then use the definition of logarithms to find the value of the expression.

Converting log6 1/36 to an Equivalent Expression, Evaluate the logarithm log6 1/36

We can use the change of base formula for logarithms to convert log6 1/36 to an equivalent expression with base 2, which is a more convenient base to work with.

log6 1/36 = log6 6^-2 =

• 2 log6 6 =
• 2(1) =
• 2

Evaluating log6 1/36

Therefore, log6 1/36 is equal to -2.

Simplifying the Result

The result can be further simplified to -2, which is the exact value of log6 1/36.

Applications of Logarithms

Logarithms have a wide range of applications in various fields, including science, engineering, and finance. They provide a convenient way to represent and manipulate data that spans several orders of magnitude, simplify complex calculations, and solve real-world problems.

In science, logarithms are used to measure the intensity of earthquakes, sound levels, and the brightness of stars. They are also essential in chemistry for calculating pH levels and equilibrium constants. In engineering, logarithms are used to design amplifiers, filters, and other electronic circuits.

In finance, logarithms are used to calculate compound interest, present value, and future value.

Solving Real-World Problems

Logarithms can be used to solve a variety of real-world problems. For example, they can be used to determine the half-life of a radioactive substance, the time it takes for an investment to double in value, or the distance to a star based on its apparent brightness.

FAQ Compilation: Evaluate The Logarithm Log6 1/36

What is the definition of a logarithm?

A logarithm is an exponent to which a base must be raised to produce a given number.

What are the properties of logarithms?

Logarithms possess several properties, including the product rule, quotient rule, power rule, and change-of-base formula.

How do you evaluate logarithms?

To evaluate logarithms, you can use the properties of logarithms or the change-of-base formula to convert the logarithm to an equivalent expression with a more convenient base.