Question

Use the change-of-base formula to evaluate the logarithm. $$\log _3 \frac{9}{40}$$

Short answer

The value of \(\log _3 \frac{9}{40}\) is approximately -1.772

Step-by-step solution

Identify the given logarithm

We are given a logarithm with a base of 3 and a value within the logarithm of \(\frac{9}{40}\). So, this can be represented as \(\log_3 \frac{9}{40}\)

Apply the change of base formula

The change of base formula can be used to express this logarithm in terms of a different base, like base 10 for example. We apply the formula, so it turns into \(\log _3 \frac{9}{40} = \frac{\log(\frac{9}{40})}{\log(3)}\)

Calculate the value of each new logarithm

Using a calculator, we can find the values of both \(\log \frac{9}{40}\) and \(\log 3\). Respectively, those values are -0.8450980400142568 and 0.47712125471966244. Substituting this into our expression gives us \(-0.8450980400142568 / 0.47712125471966244\)

Calculate the value of the fraction

Dividing -0.8450980400142568 by 0.47712125471966244 gives us -1.7719568419318752

Key concepts

Logarithms

Logarithms are a mathematical concept that helps us deal with exponential equations and simplify complex calculations. Think of a logarithm as "the reverse" of raising a number to a power. They tell us what power a base number must be raised to, to achieve another number. For example, \(\log_{10} 100 = 2\), because 10 raised to the power of 2 equals 100.
Logarithms come in different bases, like base 10 (common logarithm) or base e (natural logarithm). The expression \(\log_b x\) means "the power to which b must be raised, to get x."
Understanding the use of logarithms largely involves grasping their properties, like:

• Product Rule: \( \log_b (xy) = \log_b x + \log_b y\).
• Quotient Rule: \( \log_b \frac{x}{y} = \log_b x - \log_b y\).
• Power Rule: \( \log_b (x^c) = c \cdot \log_b x\).

These rules simplify breaking down complex numbers into more manageable parts or when converting between different logarithmic forms.

Base Conversion

The change of base formula is very useful when evaluating logarithms with different bases, especially when your calculator only supports common or natural logarithms. This formula helps you convert any logarithm into a simpler base, usually 10 or e.
The formula is given as:

• \(\log_b a = \frac{\log_c a}{\log_c b}\)

This expression implies there's a way to express logarithms in terms of another base by using a fraction of logarithms with that new base. This is especially useful for solving equations where a calculator default log function doesn't match the base you need.
It means \(\log_3 \frac{9}{40}\) turns into the ratio of \(\log \frac{9}{40}\) to \(\log 3\) when changing to base 10.
Ultimately, this makes solving problems easy and accessible since we can use calculators directly for simple, converted bases.

Mathematical Evaluation

Mathematical evaluation refers to the process of calculating or simplifying mathematical expressions to find an answer. When dealing with logarithms, evaluation often requires precise steps.
Here, once we apply the change of base formula, substituting it into our original problem, \(\log_3 \frac{9}{40}\) becomes a fraction calculation. You'll have:

• \(\frac{\log \frac{9}{40}}{\log 3}\)

With this new form, you plug the values into your calculator, knowing these steps:

• Calculate \(\log \frac{9}{40}\), yielding a negative (-0.8450980400142568) since the number is less than 1.
• Then, compute \(\log 3\), giving a positive result of 0.47712125471966244.

Your last task is to divide the two results, solving the fraction to reveal the logarithm's value: -1.7719568419318752.
These calculations help determine precise answers, showcasing a beautiful blend of algebraic manipulation and computational efficiency.