Problem 37 - Precalculus

Question

Evaluate the logarithms exactly (if possible). $$\log _{5} 3125$$

Step-by-Step Solution

Answer

The logarithm \(\log_{5}{3125}\) evaluates to 5.

Step 1: Identify the Base

The given logarithm is \(\log_{5}{3125}\), where 5 is the base of the logarithm.

Step 2: Express Number as a Power of Base

Find what power of 5 equals 3125. Since 3125 is a relatively small number, start by multiplying 5 by itself until you reach 3125:- \(5^{1} = 5\)- \(5^{2} = 25\)- \(5^{3} = 125\)- \(5^{4} = 625\)- \(5^{5} = 3125\) Therefore, \(3125 = 5^{5}\).

Step 3: Evaluate the Logarithm

Rewrite the logarithm in terms of the exponential expression you found: \(\log_{5}{3125} = \log_{5}{5^{5}}\).

Step 4: Apply Logarithmic Identity

Use the logarithmic identity \(\log_{b}{b^{n}} = n\), which states that the logarithm of a number with the same base raised to an exponent is equal to that exponent. Applying this to our expression: \(\log_{5}{5^{5}} = 5\).

Step 5: Final Result

The logarithm \(\log_{5}{3125}\) evaluates exactly to 5.

Key Concepts

base of a logarithm

The base of a logarithm is a crucial component when working with logarithms. In the expression \(log_b(x)\), \(b\) represents the base. The base dictates the exponential relationship between the numbers. For example, in \(\log_5(3125)\), the base is 5.

Understanding the base of a logarithm is similar to understanding the base in exponential functions. If we have \(\log_b(x) = y\), it means that \(b^y = x\). Therefore, in our example with a base of 5, we can interpret it as asking the question: "5 raised to what power equals 3125?"

Generally, the base must be a positive number other than 1. Common bases include 10, known as common logarithms, often used in scientific calculations, and \(e\) (approximately 2.718), known as natural logarithms frequently used in calculus.

exponents

Exponents are a fundamental concept in mathematics where a number, the base \(b\), is raised to a power, the exponent \(n\). This is written as \(b^n\), which means multiplying the base \(b\) by itself \(n\) times.

In our logarithm example, to evaluate \(\log_5(3125)\), we need to express 3125 as a power of 5. Through incremental multiplication, we determined that \(5^5 = 3125\). Here, 5 is the base and acts as a constant multiplier, while 5 is the exponent indicating the number of times 5 appears in the multiplication process.

Some key properties of exponents include:

• \(b^0 = 1\) for any non-zero \(b\).
• \(b^1 = b\).
• \(b^{m+n} = b^m \cdot b^n\).
• \((b^m)^n = b^{mn}\).

Understanding how to work with exponents is essential for manipulating logarithmic expressions easily.

logarithmic identities

Logarithmic identities serve as powerful tools for simplifying expressions and solving logarithmic equations. A fundamental identity used in evaluating the original exercise is \(\log_b(b^n) = n\). This identity implies that if the base of the logarithm is the same as the base of the exponent, you can simply take the exponent as the result.

In our exercise, we found \(3125\) as \(5^5\), so \(\log_5(3125) = \log_5(5^5) = 5\). This illustrates how the logarithmic identity allows for simplification and direct evaluation.

Other important logarithmic identities include:

• \(\log_b(mn) = \log_b(m) + \log_b(n)\)
• \(\log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n)\)
• \(\log_b(m^n) = n \cdot \log_b(m)\)

These identities enable more complex logarithmic calculations by breaking them down into more manageable parts.