Which of the following is equivalent to log9w? This question embarks us on a journey into the fascinating realm of logarithmic equivalence, where we uncover the intricate relationship between different logarithmic expressions and delve into their practical applications.

Logarithms play a pivotal role in various fields, including science, engineering, and finance, providing a powerful tool for simplifying complex calculations and solving equations. Understanding the equivalence of log9w to log3(3w) is crucial for harnessing the full potential of logarithms in these domains.

Mathematical Equivalence of Logarithms: Which Of The Following Is Equivalent To Log9w

Logarithmic equivalence refers to the relationship between logarithmic expressions with different bases. It allows us to convert from one base to another while maintaining the equality of the expression.

The logarithmic equivalence formula is:

log_ba = log _ca / log _cb

Using this formula, we can demonstrate the equivalence of log9w to log3(3w):

log9w = log3(3w) / log39

= log3(3w) / 2

= log3w + log33 / 2

= log3w + 1 / 2

Applications of Logarithms

Logarithms have numerous practical applications in various fields, including:

• Science: Modeling exponential growth and decay, such as radioactive decay or population growth.
• Engineering: Designing electronic circuits, amplifiers, and control systems.
• Finance: Calculating compound interest, loan payments, and investment returns.

The equivalence of log9w to log3(3w) can be utilized in these applications by allowing us to convert between different logarithmic bases, ensuring the accuracy and consistency of calculations.

For example, in finance, when calculating the compound interest on an investment, the equivalence allows us to convert from a different interest rate base to the base that is most convenient for the calculation.

Properties of Logarithms

Key properties of logarithms include:

• Product rule: log _b(xy) = log _bx + log _by
• Quotient rule: log _b(x/y) = log _bx – log _by
• Power rule: log _b(x ⁿ) = n log _bx

These properties can be applied to simplify expressions involving log9w:

log9(w³) = 3 log9w

log9(w/9) = log9w

• log99 = log9w
• 1

They can also be used to solve equations and inequalities involving logarithms.

Graph of the Logarithmic Function

The graph of the logarithmic function y = log9w is:

[Image of the graph of y = log9w]

It is an increasing function with a vertical asymptote at w = 0 and a horizontal asymptote at y = 0. The equivalence to log3(3w) is reflected in the graph, as the graph of log3(3w) is obtained by shifting the graph of log9w one unit to the right.

Numerical Approximation

Log9w can be numerically approximated using calculators or mathematical tables.

For example, using a calculator, we can find that log910 ≈ 1.046.

The accuracy of the approximation depends on the number of decimal places used in the calculation.

FAQ Insights

What is the logarithmic equivalence formula?

The logarithmic equivalence formula states that logb(x) = logc(x) / logc(b), where b and c are positive numbers and b ≠ 1.

How can I use the logarithmic equivalence formula to convert from one base to another?

To convert from base b to base c, use the formula logc(x) = logb(x) / logb(c).

What are the practical applications of logarithms?

Logarithms are used in various fields, including science, engineering, and finance, for simplifying complex calculations, solving equations, and modeling exponential growth and decay.