Question

Use the change-of-base formula to evaluate the logarithm. $$\log _5 13$$

Short answer

The result of \(\log _5 13\) using the change of base formula is \(\frac{\log 13}{\log 5}\).

Step-by-step solution

Identify the values of \(a\) and \(b\)

From the expression \(\log _5 13\), we can identify that \(a = 13\) and \(b = 5\). So, we are seeking a solution for \(\log _b a\) where \(b=5\) and \(a=13\).

Apply the Change of Base Formula

Using the change-of-base formula, the given expression can be transformed as follows: \(\log _b a = \frac{\log a}{\log b}\). Substituting the values \(b=5\) and \(a=13\), the expression becomes: \(\log _5 13 = \frac{\log 13}{\log 5}\). This gives you the desired value.

Key concepts

Change-of-Base Formula

The change-of-base formula is a powerful tool for evaluating logarithms, especially when working with calculators that may not support every base. Essentially, it allows us to rewrite a logarithm in terms of logs with bases we can work with, such as 10 or e. Here's how it works:

• For any positive numbers \(a\), \(b\), and \(c\), where \(b\) and \(c\) are not equal to 1, the change-of-base formula is expressed mathematically as: \[ \log_b a = \frac{\log_c a}{\log_c b} \]
• This conversion simplifies calculations because databases like the natural logarithm (\(\ln\)) and common logarithm (\(\log_{10}\)) are readily available on calculators.
• In this way, the expression \(\log_5 13\) can be calculated using the known logs: \(\frac{\log_{10} 13}{\log_{10} 5}\).

Whether using a scientific calculator or computer, you simply compute \(\log_{10}\) to find each value needed. This formula ensures you can evaluate any logarithm efficiently, even if the base is uncommon.

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions, allowing you to solve for unknown exponents. If you have the equation \(b^y = a\), the corresponding logarithmic function is \(y = \log_b a\). This reads as "\(y\) is the power you raise \(b\) to get \(a\)."

Key characteristics of logarithmic functions include:

• They are the inverse operations of exponentials: \(\log_b(a) = c\) implies \(b^c = a\).
• Logarithms with base ten, \(\log_{10}\), are called common logarithms. They simplify many scientific calculations.
• The natural logarithm, \(\ln\), is the logarithm with base \(e\) (approximately 2.718), frequently used in calculus and applications involving continuous growth.
• Graphs of logarithmic functions generally increase slowly, mirroring rapid growth seen in exponential graphs, but flipped around the line \(y = x\).

Understanding these basics helps build a comprehensive grasp of how logarithms function within mathematics and is essential when delving into higher-level math problems.

Evaluating Logarithms

Evaluating logarithms involves calculating the power to which the base of the logarithm must be raised to obtain a certain number. Here's a simple approach to finding the value of logarithms, like \(\log_5 13\):

• First, recognize that this expression asks you to determine the power needed on base \(5\) to achieve \(13\).
• Using the change-of-base formula, we convert \(\log_5 13\) to \(\frac{\log_{10} 13}{\log_{10} 5}\), making the calculation possible with a standard calculator.
• Therefore, compute both \(\log_{10} 13\) and \(\log_{10} 5\) separately, then divide the results to get the logarithm's value.

This technique is extremely useful as calculating logs in non-standard bases directly is often impractical with ordinary calculators. By understanding how to evaluate, change base, and identify the necessary power, solving complex logarithmic equations becomes much more approachable.