Question

Use the change-of-base formula to evaluate the logarithm. $$\log _6 17$$

Short answer

The value of \(\log_6 17\), after using the change of base formula, is approximately equal to \(\frac{\log_{10} 17}{\log_{10} 6}\), which can further be calculated using a calculator.

Step-by-step solution

Identifying the Values

In this problem, we're given \(\log_6 17\). So, the base (b) is 6 and the number (a) for which we are finding the logarithm is 17.

Applying the Change of Base Formula

Let's apply the change of base formula \(\log_b a = \frac{\log_c a}{\log_c b}\). We'll use the base 10 (c=10), which is also the most common base in logarithm. So, \(\log_6 17 = \frac{\log_{10} 17}{\log_{10} 6}\)

Simplifying the Result

Calculate the log base 10 of 17 and 6. These calculations can be done using a calculator. Once you've found these two values, divide them to get the value of \(\log_6 17\)

Key concepts

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions. They help answer the question: "To what power should we raise the base to get a certain number?" When we write \(\log_b a\), we're asking, "What power do we have to raise \(b\) to get \(a\)?". This relationship is crucial in understanding everything from compound interest growth to scientific scales like the Richter scale for earthquakes.
To understand the logarithmic function better, consider it as the other side of the coin to the exponential function. For example, if you know that \(2^3 = 8\), it follows that \(\log_2 8 = 3\).
Logarithmic functions have various applications across science, engineering, and even in social sciences, often simplifying complex multiplicative processes into more manageable additive operations.

Base 10 Logarithm

The base 10 logarithm, often known as the common logarithm, is specifically important because it uses the number 10 as the base. When we write \(\log_{10} x\), it's often abbreviated to simply \(\log x\).
This type of logarithm is used widely in textbooks and calculators because our number system is based on tens, making it convenient for many practical applications.
For example, the pH level in chemistry is calculated using base 10 logarithms to express hydrogen ion concentrations, and sound intensity levels are measured in decibels using base 10. This usefulness stems from the ease with which numbers can grow and shrink exponentially in the base 10 system.

• Helpful in handling very large or small numbers
• Common in scientific calculations

Evaluate Logarithms

Evaluating logarithms can initially seem challenging, but it becomes simple with practice and tools like the change of base formula. This formula allows us to calculate the logarithm when the base is something other than 10 or \(e\).
Using the change of base formula, \(\log_b a = \frac{\log_c a}{\log_c b}\), we can transform the problem into one involving base 10 or natural logarithms, which are more manageable. For instance, to evaluate \(\log_6 17\), use the formula setting both the numerator and denominator to base 10 logs: \(\log_6 17 = \frac{\log_{10} 17}{\log_{10} 6}\).
This operation reduces the problem to something a simple calculator can handle, making logarithmic evaluations straightforward and approachable.